ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 4 Issue 5 - 2016
On Estimating Flexible Weibull Parameters with Type I Progressive Interval Censoring with Random Removal Using Data of Cancerous Tumors in Blood
Afify WM*
Department of Head of Statistics, Mathematics & Insurance, Kafr El-sheikh University, Egypt
Received: September 09, 2016| Published: October 13, 2016
*Corresponding author: Afify WM, Department of Head of Statistics, Mathematics & Insurance, Kafr El-sheikh University, Faculty of Commerce, Egypt, Email:
Citation: Afify WM (2016) On Estimating Flexible Weibull Parameters with Type I Progressive Interval Censoring with Random Removal Using Data of Cancerous Tumors in Blood. Biom Biostat Int J 4(5): 00108. DOI: 10.15406/bbij.2016.04.00108

Abstract

In this paper, the maximum likelihood and the Bayes estimators of the two unknown parameters of the flexible Weibull distribution have been obtained for progressive Interval type-I censoring scheme with binomial random removal. Point estimation and confidence intervals based on maximum likelihood and bootstrap method are also proposed. A Bayesian approach using Markov chain Monte Carlo (MCMC) method to generate from the posterior distributions and in turn computing the Bayes estimators are developed. To illustrate the proposed methods will discuss an example with the real data. Finally, comparing the two techniques through comparisons between the maximum likelihood using bootstrap method and different Bayes estimators using MCMC study.

Keywords: Flexible Weibull distribution; Progressive interval type-I censoring; Random removal; Percentile bootstrap; Bayesian and non-Bayesian approach; Markov chain Monte Carlo (MCMC)

Introduction

Censoring is very common in life tests in the past several decades; the experimenter may be unable to obtain complete information on failure times of all experimental items. For this reason, Aggarwalla [1] suggested a useful type of censoring, namely, a progressively Type I interval censored data, which is a union of Type I interval and progressive censoring. This method of lifetime data collection can be useful to a biological experimenter, particularly when the experimental units are humans, as continuous monitoring is often not possible to implement, and withdrawal rates from such studies may high.

In progressive censored the number of units being removed from the test at each failure time may occur at random. For example; the number of patients who drop out of clinical test at each stage is random and cannot be predetermined. That is why to display a more general censoring scheme called progressive progressively Type I interval censored with random removal. It can be described as follows: suppose n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36E9@ units are put on life test at time T 0 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaBaaajuaibaGaaGimaaqcfayabaGaeyypa0JaaGimaaaa@3AB4@ and under inspection at m pre-specified times T 1 < T 2 <...< T m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaBaaajuaibaGaaGymaaqcfayabaGaeyipaWJaamivamaaBaaajuai baGaaGOmaaqcfayabaGaeyipaWJaaiOlaiaac6cacaGGUaGaeyipaW JaamivamaaBaaajuaibaGaamyBaaqcfayabaaaaa@4331@ where T m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaBaaajuaibaGaamyBaaqcfayabaaaaa@392C@ is scheduled time to terminate the experiment. The number, k i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Aam aaBaaajuaibaGaamyAaaqcfayabaaaaa@393F@ , of failures within ( T i1 , T i ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaiikai aadsfadaWgaaqcfasaaiaadMgacqGHsislcaaIXaaajuaGbeaacaGG SaGaamivamaaBaaajuaibaGaamyAaaqcfayabaGaaiyxaaaa@3FB1@ is recorded and r i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOCam aaBaaajuaibaGaamyAaaqcfayabaaaaa@3946@ surviving items are randomly removed from the life testing at the ith inspection time, T i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaBaaajuaibaGaamyAaaqcfayabaaaaa@3928@ , for i=1,2,...m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyAai abg2da9iaaigdacaGGSaGaaGOmaiaacYcacaGGUaGaaiOlaiaac6ca caWGTbaaaa@3E57@ . Since the number, Y i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamywam aaBaaajuaibaGaamyAaaqcfayabaaaaa@392D@ , of surviving items is a random variable and exact number of items with drawn should not be greater than Y i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamywam aaBaaajuaibaGaamyAaaqcfayabaaaaa@392D@ at time schedule T i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaBaaajuaibaGaamyAaaqcfayabaaaaa@3928@ , r i 's MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOCam aaBaaajuaibaGaamyAaaqabaGaai4jaiaadohaaaa@3A5B@ are random. Such a censoring mechanism is termed as progressive interval type-I censoring with random removal scheme. If we assume that probability of removal of a unit at every stage is π for each unit then ri can be considered to follow a binomial distribution i.e, r i B( nm j=0 i1 r j ,π ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOCam aaBaaajuaibaGaamyAaaqcfayabaGaeyisISRaamOqamaabmaabaGa amOBaiabgkHiTiaad2gacqGHsisldaaeWaqaaiaadkhadaWgaaqcfa saaiaadQgaaKqbagqaaiaacYcacqaHapaCaKqbGeaacaWGQbGaeyyp a0JaaGimaaqaaiaadMgacqGHsislcaaIXaaajuaGcqGHris5aaGaay jkaiaawMcaaaaa@4E21@ for i=1,2,...,m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyAai abg2da9iaaigdacaGGSaGaaGOmaiaacYcacaGGUaGaaiOlaiaac6ca caGGSaGaamyBaaaa@3F07@ . The main difference between progressive interval type I censoring with fixed removal and progressive interval type I censoring with random removals is that the removals are predetermined in the former case while they are random in the latter case. Note that m is pre-determined in both cases. However, many practical applications suggest that it is more flexible to have removals random to accommodate the unexpected drop out of experimental subjects.

Although progressive censoring occurs frequently in many applications, there are relatively few works on it. Some early works can be found in Cohen [2], Readers can refer to the book Balakrishnan & Aggarwala [3] for more details on the methods and applications of this topic. However, all these works assumed that the number of units being removed from the test is fixed in advance. In practice, it is impossible to pre-determine the removal pattern. Thus, Yuen & Tse [4] and Yang et al. [5] considered the estimation problem when lifetimes collected under a Type II progressive censoring with random removals and Kendell & Anderson [6] point out that the expected duration under grouped data. Progressive type-I interval censored sampling is an important practical problem that has received considerable attention in the past several years. Based on the progressive type-I interval censored sampling, Ashour & Afify [7] derived the maximum likelihood estimators of parameters of the exponentiated Weibull family and their asymptotic variances under random removal. Lin et al. [8] determined optimally spaced inspection times for the log-normal distribution, while Ng & Wang [9] and Chen & Lio [10] compared three classical estimation methods, the maximum likelihood estimators the moment method and the probability plot method in terms of the Weibull distribution and generalized exponential respectively.

In Bayesian approach, It is too difficult to find integrate over the posterior distribution and the problem is that the integrals are usually impossible to evaluate analytically. But in MCMC technique, the MCMC methodology provided a convenient and efficient way to sample from complex, high-dimensional statistical distributions. Recently, application of the MCMC method to the estimation of parameters or some other vital properties about statistical models is very common. Green et al. [11] using the MCMC method for estimating the three parameters Weibull distribution, and they showed that the MCMC method is better than the ML method, when given a proper prior distribution of the parameters. As a generalization of the two parameter Weibull model, Gupta et al. [12] gave a complete Bayesian analysis of the Weibull extension model using MCMC simulation and complete sample. Lin & Lio [13] discussed Bayesian inference under progressive type I interval censoring by using MCMC.

A random variable x is said to have a Flexible Weibull Distribution with parameters λ,β>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW Maaiilaiabek7aIjabg6da+iaaicdaaaa@3C4B@ if its probability density function, cumulative function, survival function and hazard function are given by

f( x;λ,β )=( λ+ β x 2 ) e λx β x e λx β x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamiEaiaacUdacqaH7oaBcaGGSaGaeqOSdigacaGLOaGa ayzkaaGaeyypa0ZaaeWaaeaacqaH7oaBcqGHRaWkdaWcaaqaaiabek 7aIbqaaiaadIhadaahaaqabKqbGeaacaaIYaaaaaaaaKqbakaawIca caGLPaaacaWGLbWaaWbaaeqajuaibaGaeq4UdWMaamiEaiabgkHiTK qbaoaalaaajuaibaGaeqOSdigabaGaamiEaaaaaaqcfaOaamyzamaa CaaabeqcfasaaiabeU7aSjaadIhacqGHsisljuaGdaWcaaqcfasaai abek7aIbqaaiaadIhaaaaaaaaa@5938@ (1)

F( x;λ,β )=1 e e λx β x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aabmaabaGaamiEaiaacUdacqaH7oaBcaGGSaGaeqOSdigacaGLOaGa ayzkaaGaeyypa0JaaGymaiabgkHiTiaadwgadaahaaqabKqbGeaacq GHsislcaWGLbqcfa4aaWbaaKqbGeqabaGaeq4UdWMaamiEaiabgkHi TKqbaoaalaaajuaibaGaeqOSdigabaGaamiEaaaaaaaaaaaa@4C3E@ (2)

F ¯ ( x;λ,β )= e e λx β x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmOray aaraWaaeWaaeaacaWG4bGaai4oaiabeU7aSjaacYcacqaHYoGyaiaa wIcacaGLPaaacqGH9aqpcaWGLbWaaWbaaeqajuaibaGaeyOeI0Iaam yzaKqbaoaaCaaajuaibeqaaiabeU7aSjaadIhacqGHsisljuaGdaWc aaqcfasaaiabek7aIbqaaiaadIhaaaaaaaaaaaa@4AAE@ (3) respectively.

In this paper we consider the Bayesian inference of the scale parameters for progressive interval type-I censored data when both parameters are unknown. We assumed that the both scale parameters λ and β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW geaaaaaaaaa8qacaGGGcGaamyyaiaad6gacaWGKbGaaiiOaiabek7a Ibaa@3F03@ have gamma prior and they are independently distributed. As expected in this case also, the Bayes estimates cannot be obtained in closed form. We propose to use the Gibbs sampling procedure to generate MCMC samples, and then using the Metropolis–Hastings algorithms, we obtain the Bayes estimates of the unknown parameters. We perform some simulation experiments to see the behavior of the proposed Bayes estimators and compare their performances with the maximum likelihood estimators.

The rest of the paper is organized as follows. In the next section, the ML estimators of the unknown parameters and approximate confidence intervals are presented. The corresponding parametric bootstrap confidence intervals for the parameters are given in Section 3. In Section 4, we cover Bayes estimates and construction of credible intervals using the MCMC techniques. In Section 5, for illustrative purposes, we performed a real data analysis. Comparisons among estimators are investigated through Monte Carlo simulations in Section 6. Finally, conclusions appear in Section 7.

Classical Estimation and Percentile Bootstrap Algorithm (Boot-p)

Classical estimation (maximum likelihood estimators) of the unknown parameters and approximate confidence intervals are presented. Also, the corresponding parametric bootstrap confidence intervals using percentile bootstrap Algorithm (Boot-p) for the parameters are given in this section.

Classical estimation

Suppose a progressively Type-I interval censored sample is collected as described above, beginning with a random sample of units with a continuous lifetime distribution F(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOrai aacIcacaWG4bGaaiykaaaa@39A5@ and let k 1 , k 2 ,..., k m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Aam aaBaaajuaibaGaaGymaaqcfayabaGaaiilaiaadUgadaWgaaqcfasa aiaaikdaaKqbagqaaiaacYcacaGGUaGaaiOlaiaac6cacaGGSaGaam 4AamaaBaaajuaibaGaamyBaaqcfayabaaaaa@427A@ denote the number of units known to have failed in the intervals (0, T 1 ],( T 1 , T 2 ],...,( T m1 , T m ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaiikai aaicdacaGGSaGaamivamaaBaaajuaibaGaaGymaaqcfayabaGaaiyx aiaacYcacaGGOaGaamivamaaBaaajuaibaGaaGymaaqcfayabaGaai ilaiaadsfadaWgaaqcfasaaiaaikdaaKqbagqaaiaac2facaGGSaGa aiOlaiaac6cacaGGUaGaaiilaiaacIcacaWGubWaaSbaaKqbGeaaca WGTbGaeyOeI0IaaGymaaqcfayabaGaaiilaiaadsfadaWgaaqcfasa aiaad2gaaKqbagqaaiaac2faaaa@5067@ , respectively. Then, based on this observed data, the joint likelihood function will be Aggarwala [1].

L 1 ( X;λ,β\R )=C i=1 m [ F( T i ;λ,β)F( T i1 ;λ,β) ] k i [ 1F( T i ;λ,β) ] r i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitam aaBaaajuaibaGaaGymaaqcfayabaWaaeWaaeaacaWGybGaai4oaiab eU7aSjaacYcacqaHYoGycaGGCbGaamOuaaGaayjkaiaawMcaaiabg2 da9iaadoeadaqeWbqaamaadmaabaGaamOraiaacIcacaWGubWaaSba aKqbGeaacaWGPbaajuaGbeaacaGG7aGaeq4UdWMaaiilaiabek7aIj aacMcacqGHsislcaWGgbGaaiikaiaadsfadaWgaaqcfasaaiaadMga cqGHsislcaaIXaaajuaGbeaacaGG7aGaeq4UdWMaaiilaiabek7aIj aacMcaaiaawUfacaGLDbaadaahaaqabKqbGeaacaWGRbqcfa4aaSba aKqbGeaacaWGPbaabeaaaaqcfa4aamWaaeaacaaIXaGaeyOeI0Iaam OraiaacIcacaWGubWaaSbaaKqbGeaacaWGPbaajuaGbeaacaGG7aGa eq4UdWMaaiilaiabek7aIjaacMcaaiaawUfacaGLDbaadaahaaqabK qbGeaacaWGYbqcfa4aaSbaaKqbGeaacaWGPbaabeaaaaaabaGaamyA aiabg2da9iaaigdaaeaacaWGTbaajuaGcqGHpis1aaaa@7554@ (4)

Where C is constant. Clearly, if r i =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOCam aaBaaajuaibaGaamyAaaqcfayabaGaeyypa0JaaGimaaaa@3B06@ for i=1,2,...,m1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyAai abg2da9iaaigdacaGGSaGaaGOmaiaacYcacaGGUaGaaiOlaiaac6ca caGGSaGaamyBaiabgkHiTiaaigdaaaa@40AF@ and r m =nk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOCam aaBaaajuaibaGaamyBaaqcfayabaGaeyypa0JaamOBaiabgkHiTiaa dUgaaaa@3D20@ equation (4) reduces to the likelihood function for interval type I censoring data is defined as follows:

L(X;λ,β)=C (1F( T m ,λ,β)) nk i=1 m (F( T i ,λ,β)F( T i1 ,λ,β)) k i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitai aacIcacaWGybGaai4oaiabeU7aSjaacYcacqaHYoGycaGGPaGaeyyp a0Jaam4qaiaacIcacaaIXaGaeyOeI0IaamOraiaacIcacaWGubWaaS baaKqbGeaacaWGTbaajuaGbeaacaGGSaGaeq4UdWMaaiilaiabek7a IjaacMcacaGGPaWaaWbaaeqajuaibaGaamOBaiabgkHiTiaadUgaaa qcfa4aaebCaeaacaGGOaGaamOraiaacIcacaWGubWaaSbaaKqbGeaa caWGPbaajuaGbeaacaGGSaGaeq4UdWMaaiilaiabek7aIjaacMcacq GHsislcaWGgbGaaiikaiaadsfadaWgaaqcfasaaiaadMgacqGHsisl caaIXaaajuaGbeaacaGGSaGaeq4UdWMaaiilaiabek7aIjaacMcaca GGPaWaaWbaaeqajuaibaGaam4AaKqbaoaaBaaajuaibaGaamyAaaqa baaaaaqaaiaadMgacqGH9aqpcaaIXaaabaGaamyBaaqcfaOaey4dIu naaaa@7088@

Where k= i=1 m k i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Aai abg2da9maaqahabaGaam4AamaaBaaajuaibaGaamyAaaqcfayabaaa juaibaGaamyAaiabg2da9iaaigdaaeaacaWGTbaajuaGcqGHris5aa aa@41BC@ and k 1 , k 2 ,..., k m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Aam aaBaaajuaibaGaaGymaaqcfayabaGaaiilaiaadUgadaWgaaqcfasa aiaaikdaaKqbagqaaiaacYcacaGGUaGaaiOlaiaac6cacaGGSaGaam 4AamaaBaaajuaibaGaamyBaaqcfayabaaaaa@427A@ are the number of units known to have failed in the intervals (0, T 1 ],( T 1 , T 2 ],...,( T m1 , T m ], MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaiikai aaicdacaGGSaGaamivamaaBaaajuaibaGaaGymaaqcfayabaGaaiyx aiaacYcacaGGOaGaamivamaaBaaajuaibaGaaGymaaqcfayabaGaai ilaiaadsfadaWgaaqcfasaaiaaikdaaKqbagqaaiaac2facaGGSaGa aiOlaiaac6cacaGGUaGaaiilaiaacIcacaWGubWaaSbaaKqbGeaaca WGTbGaeyOeI0IaaGymaaqcfayabaGaaiilaiaadsfadaWgaaqcfasa aiaad2gaaKqbagqaaiaac2facaGGSaaaaa@5117@ respectively.

For type I progressive Interval censoring, supposed that r i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGPbaabeaaaaa@3807@ is independent of X i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwam aaBaaajuaibaGaamyAaaqabaaaaa@389E@ for all i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyAaa aa@3772@ ; Wu & Chang [14] suggested the following likelihood function of a progressive interval censoring with binomial removals

L(X,R;λ,β)= L 1 (X;λ,β\R=r)×P(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitai aacIcacaWGybGaaiilaiaadkfacaGG7aGaeq4UdWMaaiilaiabek7a IjaacMcacqGH9aqpcaWGmbWaaSbaaKqbGeaacaaIXaaajuaGbeaaca GGOaGaamiwaiaacUdacqaH7oaBcaGGSaGaeqOSdiMaaiixaiaadkfa cqGH9aqpcaWGYbGaaiykaiabgEna0kaadcfacaGGOaGaamOuaiaacM caaaa@530F@ (5)

Where L 1 (X;θ\R=r) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitam aaBaaajuaibaGaaGymaaqcfayabaGaaiikaiaadIfacaGG7aGaeqiU deNaaiixaiaadkfacqGH9aqpcaWGYbGaaiykaaaa@414C@ is the likelihood function for a progressive type I interval censored with fixed removal (4) and P(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiuai aacIcacaWGsbGaaiykaaaa@3989@ will be

P(R)=P( R m1 = r m1 \ R m2 = r m2 ,..., R 1 = r 1 )P( R m2 = r m2 \ R m3 = r m3 ,..., R 1 = r 1 )... P( R 2 = r 2 \ R 1 = r 1 )P( R 1 = r 1 )....P( R 2 = r 2 \ R 1 = r 1 )P( R 1 = r 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGca WGqbGaaiikaiaadkfacaGGPaGaeyypa0JaamiuaiaacIcacaWGsbWa aSbaaKqbGeaacaWGTbGaeyOeI0IaaGymaaqcfayabaGaeyypa0Jaam OCamaaBaaajuaibaGaamyBaiabgkHiTiaaigdaaKqbagqaaiaacYfa caWGsbWaaSbaaKqbGeaacaWGTbGaeyOeI0IaaGOmaaqcfayabaGaey ypa0JaamOCamaaBaaajuaibaGaamyBaiabgkHiTiaaikdaaKqbagqa aiaacYcacaGGUaGaaiOlaiaac6cacaGGSaGaamOuamaaBaaajuaiba GaaGymaaqcfayabaGaeyypa0JaamOCamaaBaaajuaibaGaaGymaaqc fayabaGaaiykaiaadcfacaGGOaGaamOuamaaBaaajuaibaGaamyBai abgkHiTiaaikdaaKqbagqaaiabg2da9iaadkhadaWgaaqcfasaaiaa d2gacqGHsislcaaIYaaajuaGbeaacaGGCbGaamOuamaaBaaajuaiba GaamyBaiabgkHiTiaaiodaaKqbagqaaiabg2da9iaadkhadaWgaaqc fasaaiaad2gacqGHsislcaaIZaaajuaGbeaacaGGSaGaaiOlaiaac6 cacaGGUaGaaiilaiaadkfadaWgaaqcfasaaiaaigdaaKqbagqaaiab g2da9iaadkhadaWgaaqcfasaaiaaigdaaKqbagqaaiaacMcacaGGUa GaaiOlaiaac6caaOqaaKqbakaadcfacaGGOaGaamOuamaaBaaajuai baGaaGOmaaqcfayabaGaeyypa0JaamOCamaaBaaajuaibaGaaGOmaa qcfayabaGaaiixaiaadkfadaWgaaqcfasaaiaaigdaaKqbagqaaiab g2da9iaadkhadaWgaaqcfasaaiaaigdaaKqbagqaaiaacMcacaWGqb GaaiikaiaadkfadaWgaaqcfasaaiaaigdaaKqbagqaaiabg2da9iaa dkhadaWgaaqcfasaaiaaigdaaKqbagqaaiaacMcacaGGUaGaaiOlai aac6cacaGGUaGaamiuaiaacIcacaWGsbWaaSbaaKqbGeaacaaIYaaa juaGbeaacqGH9aqpcaWGYbWaaSbaaKqbGeaacaaIYaaajuaGbeaaca GGCbGaamOuamaaBaaajuaibaGaaGymaaqcfayabaGaeyypa0JaamOC amaaBaaajuaibaGaaGymaaqcfayabaGaaiykaiaadcfacaGGOaGaam OuamaaBaaajuaibaGaaGymaaqcfayabaGaeyypa0JaamOCamaaBaaa juaibaGaaGymaaqcfayabaGaaiykaaaaaa@AEC0@

Such as

P(R)= (nm)! i=1 m r i !(nm j=1 m1 r j )! π j=1 m1 r j (1π) (m1)(nm) j=1 m1 (mj) r j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiuai aacIcacaWGsbGaaiykaiabg2da9maalaaabaGaaiikaiaad6gacqGH sislcaWGTbGaaiykaiaacgcaaeaadaqeWbqaaiaadkhadaWgaaqcfa saaiaadMgaaKqbagqaaiaacgcacaGGOaGaamOBaiabgkHiTiaad2ga cqGHsisldaaeWbqaaiaadkhadaWgaaqcfasaaiaadQgaaKqbagqaai aacMcacaGGHaaajuaibaGaamOAaiabg2da9iaaigdaaeaacaWGTbGa eyOeI0IaaGymaaqcfaOaeyyeIuoaaKqbGeaacaWGPbGaeyypa0JaaG ymaaqaaiaad2gaaKqbakabg+Givdaaaiabec8aWnaaCaaabeqaamaa qahabaGaamOCamaaBaaajuaibaGaamOAaaqcfayabaaajuaibaGaam OAaiabg2da9iaaigdaaeaacaWGTbGaeyOeI0IaaGymaaqcfaOaeyye IuoaaaGaaiikaiaaigdacqGHsislcqaHapaCcaGGPaWaaWbaaeqaju aibaGaaiikaiaad2gacqGHsislcaaIXaGaaiykaiaacIcacaWGUbGa eyOeI0IaamyBaiaacMcacqGHsisljuaGdaaeWbqcfasaaiaacIcaca WGTbGaeyOeI0IaamOAaiaacMcacaWGYbqcfa4aaSbaaKqbGeaacaWG QbaabeaaaeaacaWGQbGaeyypa0JaaGymaaqaaiaad2gacqGHsislca aIXaaacqGHris5aaaaaaa@8401@ (6)

and f( . ),F( . ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaaiOlaaGaayjkaiaawMcaaiaacYcacaWGgbWaaeWaaeaa caGGUaaacaGLOaGaayzkaaaaaa@3D60@ are the same as defined before in (1) and (2) respectively. The log likelihood function with random removal can be written as

logL(X,R;λ,β)=logC+ i=1 m k i log[ F( T i )F( T i1 ) ]+ i=1 m r i log[ 1F( T i ) ] + j=1 m1 r j logπ+(m1)(nm) j=1 m1 (mj) r j log(1π) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGci GGSbGaai4BaiaacEgacaWGmbGaaiikaiaadIfacaGGSaGaamOuaiaa cUdacqaH7oaBcaGGSaGaeqOSdiMaaiykaiabg2da9iGacYgacaGGVb Gaai4zaiaadoeacqGHRaWkdaaeWbqaaiaadUgadaWgaaqcfasaaiaa dMgaaKqbagqaaiGacYgacaGGVbGaai4zamaadmaabaGaamOraiaacI cacaWGubWaaSbaaKqbGeaacaWGPbaajuaGbeaacaGGPaGaeyOeI0Ia amOraiaacIcacaWGubWaaSbaaKqbGeaacaWGPbGaeyOeI0IaaGymaa qcfayabaGaaiykaaGaay5waiaaw2faaiabgUcaRmaaqahabaGaamOC amaaBaaajuaibaGaamyAaaqcfayabaGaciiBaiaac+gacaGGNbWaam WaaeaacaaIXaGaeyOeI0IaamOraiaacIcacaWGubWaaSbaaKqbGeaa caWGPbaajuaGbeaacaGGPaaacaGLBbGaayzxaaaajuaibaGaamyAai abg2da9iaaigdaaeaacaWGTbaajuaGcqGHris5aaqcfasaaiaadMga cqGH9aqpcaaIXaaabaGaamyBaaqcfaOaeyyeIuoacqGHRaWkaeaada aeWbqaaiaadkhadaWgaaqcfasaaiaadQgaaKqbagqaaaqcfasaaiaa dQgacqGH9aqpcaaIXaaabaGaamyBaiabgkHiTiaaigdaaKqbakabgg HiLdGaciiBaiaac+gacaGGNbGaeqiWdaNaey4kaSIaaiikaiaad2ga cqGHsislcaaIXaGaaiykaiaacIcacaWGUbGaeyOeI0IaamyBaiaacM cacqGHsisldaaeWbqaaiaacIcacaWGTbGaeyOeI0IaamOAaiaacMca caWGYbWaaSbaaKqbGeaacaWGQbaajuaGbeaaaKqbGeaacaWGQbGaey ypa0JaaGymaaqaaiaad2gacqGHsislcaaIXaaajuaGcqGHris5aiGa cYgacaGGVbGaai4zaiaacIcacaaIXaGaeyOeI0IaeqiWdaNaaiykaa aaaa@A8AA@ (7)

The maximum likelihood estimations of λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW gaaa@3838@ and β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@ are the simultaneous solutions of following normal equations

i=1 m k i F( T i ) λ F( T i1 ) λ F( T i )F( T i1 ) + i=1 m r i λ [ 1F( T i ) ] [ 1F( T i ) ] =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaabmae aacaWGRbWaaSbaaKqbGeaacaWGPbaajuaGbeaaaKqbGeaacaWGPbGa eyypa0JaaGymaaqaaiaad2gaaKqbakabggHiLdWaaSaaaeaadaWcaa qaaiabgkGi2kaadAeadaqadaqaaiaadsfadaWgaaqcfasaaiaadMga aKqbagqaaaGaayjkaiaawMcaaaqaaiabgkGi2kabeU7aSbaacqGHsi sldaWcaaqaaiabgkGi2kaadAeadaqadaqaaiaadsfadaWgaaqcfasa aiaadMgacqGHsislcaaIXaaajuaGbeaaaiaawIcacaGLPaaaaeaacq GHciITcqaH7oaBaaaabaGaamOramaabmaabaGaamivamaaBaaajuai baGaamyAaaqcfayabaaacaGLOaGaayzkaaGaeyOeI0IaamOramaabm aabaGaamivamaaBaaajuaibaGaamyAaiabgkHiTiaaigdaaKqbagqa aaGaayjkaiaawMcaaaaacqGHRaWkdaaeWaqaaiaadkhadaWgaaqcfa saaiaadMgaaKqbagqaaaqcfasaaiaadMgacqGH9aqpcaaIXaaabaGa amyBaaqcfaOaeyyeIuoadaWcaaqaamaalaaabaGaeyOaIylabaGaey OaIyRaeq4UdWgaamaadmaabaGaaGymaiabgkHiTiaadAeadaqadaqa aiaadsfadaWgaaqcfasaaiaadMgaaKqbagqaaaGaayjkaiaawMcaaa Gaay5waiaaw2faaaqaamaadmaabaGaaGymaiabgkHiTiaadAeadaqa daqaaiaadsfadaWgaaqcfasaaiaadMgaaKqbagqaaaGaayjkaiaawM caaaGaay5waiaaw2faaaaacqGH9aqpcaaIWaaaaa@832F@ (8)

i=1 m k i F( T i ) β F( T i1 ) β F( T i )F( T i1 ) + i=1 m r i β [ 1F( T i ) ] [ 1F( T i ) ] =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaGfWbqabKqbG8aabaWdbiaadMgacqGH9aqpcaaIXaaapaqa a8qacaWGTbaajuaGpaqaa8qacqGHris5aaGaam4Aa8aadaWgaaqcfa saa8qacaWGPbaajuaGpaqabaWdbmaalaaapaqaa8qadaWcaaWdaeaa peGaeyOaIyRaamOramaabmaapaqaa8qacaWGubWdamaaBaaajuaiba WdbiaadMgaaKqba+aabeaaa8qacaGLOaGaayzkaaaapaqaa8qacqGH ciITcqaHYoGyaaGaeyOeI0YaaSaaa8aabaWdbiabgkGi2kaadAeada qadaWdaeaapeGaamiva8aadaWgaaqcfasaa8qacaWGPbGaeyOeI0Ia aGymaaqcfa4daeqaaaWdbiaawIcacaGLPaaaa8aabaWdbiabgkGi2k abek7aIbaaa8aabaWdbiaadAeadaqadaWdaeaapeGaamiva8aadaWg aaqcfasaa8qacaWGPbaajuaGpaqabaaapeGaayjkaiaawMcaaiabgk HiTiaadAeadaqadaWdaeaapeGaamiva8aadaWgaaqcfasaa8qacaWG PbGaeyOeI0IaaGymaaqcfa4daeqaaaWdbiaawIcacaGLPaaaaaGaey 4kaSYaaybCaeqajuaipaqaa8qacaWGPbGaeyypa0JaaGymaaWdaeaa peGaamyBaaqcfa4daeaapeGaeyyeIuoaaiaadkhapaWaaSbaaKqbGe aapeGaamyAaaqcfa4daeqaa8qadaWcaaWdaeaapeWaaSaaa8aabaWd biabgkGi2cWdaeaapeGaeyOaIyRaeqOSdigaamaadmaapaqaa8qaca aIXaGaeyOeI0IaamOramaabmaapaqaa8qacaWGubWdamaaBaaajuai baWdbiaadMgaaKqba+aabeaaa8qacaGLOaGaayzkaaaacaGLBbGaay zxaaaapaqaa8qadaWadaWdaeaapeGaaGymaiabgkHiTiaadAeadaqa daWdaeaapeGaamiva8aadaWgaaqcfasaa8qacaWGPbaajuaGpaqaba aapeGaayjkaiaawMcaaaGaay5waiaaw2faaaaacqGH9aqpcaaIWaaa aa@887F@ (9)

Note that P(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiuai aacIcacaWGsbGaaiykaaaa@3989@ does not involve the parameters. Therefore, the MLE π ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiWdaNbaK aaaaa@37C3@ of π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiWdahaaa@37B3@ can be found by maximizing P(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiuai aacIcacaWGsbGaaiykaaaa@3989@ directly, that is,

1 π j=1 m1 r j 1 1 π ( (m1)(nm) j=1 m1 (mj) r j ) =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaaGymaaqaaiqbec8aWzaataaaamaaqahabaGaamOCamaaBaaajuai baGaamOAaaqcfayabaGaeyOeI0YaaSaaaeaacaaIXaaabaGaaGymai abgkHiTiqbec8aWzaataaaamaabmaabaGaaiikaiaad2gacqGHsisl caaIXaGaaiykaiaacIcacaWGUbGaeyOeI0IaamyBaiaacMcacqGHsi sldaaeWbqaaiaacIcacaWGTbGaeyOeI0IaamOAaiaacMcacaWGYbWa aSbaaKqbGeaacaWGQbaajuaGbeaaaKqbGeaacaWGQbGaeyypa0JaaG ymaaqaaiaad2gacqGHsislcaaIXaaajuaGcqGHris5aaGaayjkaiaa wMcaaaqcfasaaiaadQgacqGH9aqpcaaIXaaabaGaamyBaiabgkHiTi aaigdaaKqbakabggHiLdGaeyypa0JaaGimaaaa@64B3@

Therefore, the maximum likelihood estimation of parameter π ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiWdaNbaK aaaaa@37C3@ is given by

π ^ = j=1 m1 r j (m1)(nm) j=1 m1 (mj1) r j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbec8aWz aajaGaeyypa0ZaaSaaaeaadaaeWbqaaiaadkhadaWgaaqcfasaaiaa dQgaaKqbagqaaaqcfasaaiaadQgacqGH9aqpcaaIXaaabaGaamyBai abgkHiTiaaigdaaKqbakabggHiLdaabaGaaiikaiaad2gacqGHsisl caaIXaGaaiykaiaacIcacaWGUbGaeyOeI0IaamyBaiaacMcacqGHsi sldaaeWbqaaiaacIcacaWGTbGaeyOeI0IaamOAaiabgkHiTiaaigda caGGPaGaamOCamaaBaaajuaibaGaamOAaaqcfayabaaajuaibaGaam OAaiabg2da9iaaigdaaeaacaWGTbGaeyOeI0IaaGymaaqcfaOaeyye Iuoaaaaaaa@5E1C@ (10)

It may be noted that (9) and (10) cannot be solved simultaneously to provide a nicely closed form for the estimators. Therefore, we propose to use fixed point iteration method for solving these equations. Using Fisher information matrix I( λ ^ , β ^ , π ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMeaca GGOaGafq4UdWMbaKaacaGGSaGafqOSdiMbaKaacaGGSaGafqiWdaNb ambacaGGPaaaaa@3F4C@ in the Appendix and the asymptotic normality of the maximum likelihood estimators can be used to compute the approximate confidence intervals (ACI) for parameters λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSb aa@382D@ , β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIb aa@381A@ and π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@ Therefore, ( 1γ )100% MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaaIXaGaeyOeI0Iaeq4SdCgacaGLOaGaayzkaaGaaGymaiaaicda caaIWaGaaiyjaaaa@3E35@ confidence intervals for parameters λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSb aa@382D@ , β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIb aa@381A@ and π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@ will be become

λ ^ ± Z γ/2 Var( α ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacuaH7oaBgaqcaiabgglaXkaadQfapaWaaSbaaKqbGeaapeGa eq4SdCMaai4laiaaikdaaKqba+aabeaapeWaaOaaa8aabaWdbiaadA facaWGHbGaamOCamaabmaapaqaaiqbeg7aHzaajaaapeGaayjkaiaa wMcaaaqabaaaaa@45A5@ , β ^ ± Z γ/2 Var( β ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacuaHYoGygaqcaiabgglaXkaadQfapaWaaSbaaKqbGeaapeGa eq4SdCMaai4laiaaikdaaKqba+aabeaapeWaaOaaa8aabaWdbiaadA facaWGHbGaamOCamaabmaapaqaaiqbek7aIzaajaaapeGaayjkaiaa wMcaaaqabaaaaa@4594@ and π ^ ± Z γ/2 Var( π ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacuaHapaCgaqcaiabgglaXkaadQfapaWaaSbaaKqbGeaapeGa eq4SdCMaai4laiaaikdaaKqba+aabeaapeWaaOaaa8aabaWdbiaadA facaWGHbGaamOCamaabmaapaqaaiqbec8aWzaajaaapeGaayjkaiaa wMcaaaqabaaaaa@45CC@

Where Z γ/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGAbWdamaaBaaajuaibaWdbiaabo7acaGGVaGaaGOmaaqc fa4daeqaaaaa@3B35@ is percentile of the standard normal distribution with right-tail probability  γ/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGGcGaae4Sdiaac+cacaaIYaaaaa@3A70@ .

Data algorithm

The data generation is based on the algorithm proposed by Aggarwala [1] to simulate the numbers,   k i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaGGGcGaam4Aa8aadaWgaaqcfasaa8qacaWGPbaajuaGpaqa baaaaa@3AB2@ of failed items in each subinterval ( T i1 , T i ],i=1,,m, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaqcWaWdaeaapeGaamiva8aadaWgaaqcfasaa8qacaWGPbGa eyOeI0IaaGymaaqcfa4daeqaa8qacaGGSaGaamiva8aadaWgaaqcfa saa8qacaWGPbaajuaGpaqabaaapeGaayjkaiaaw2faaiaacYcacaqG PbGaeyypa0JaaGymaiaacYcacqGHMacVcaGGSaGaaeyBaiaacYcaaa a@48BD@ from an initial sample of size putting on life testing at time 0. This algorithm, which is an extension from the procedure developed by Kemp & Kemp [15] for the multinomial distribution, involves generating m binomial random variables. A procedure to generate a progressively type I interval censored data with random removal, ( k i ,  r i ,  T i ), i=1,,m, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaGGOaGaam4Aa8aadaWgaaqcfasaa8qacaWGPbaajuaGpaqa baWdbiaacYcacaqGGcGaamOCa8aadaWgaaqcfasaa8qacaWGPbaaju aGpaqabaWdbiaacYcacaqGGcGaamiva8aadaWgaaqcfasaa8qacaWG PbaajuaGpaqabaWdbiaacMcacaGGSaGaaiiOaiaadMgacqGH9aqpca aIXaGaaiilaiabgAci8kaacYcacaWGTbGaaiilaaaa@4D92@ from the flexible Weibull distribution can be described as follows briefly: let k 0 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGRbWdamaaBaaajuaibaWdbiaaicdaaKqba+aabeaapeGa eyypa0JaaGimaaaa@3B2A@ and r 0 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGYbWdamaaBaaajuaibaWdbiaaicdaaKqba+aabeaapeGa eyypa0JaaGimaaaa@3B31@ and for i=1,,m, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGPbGaeyypa0JaaGymaiaacYcacqGHMacVcaGGSaGaamyB aiaacYcaaaa@3DE4@

Step 1: set i=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyAai abg2da9iaaicdaaaa@3933@ and let k sum=r sum=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGRbGaaiiOaiaadohacaWG1bGaamyBaiabg2da9iaadkha caGGGcGaam4CaiaadwhacaWGTbGaeyypa0JaaGimaaaa@4362@ .

Step 2:  i=i+1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcfaieaa aaaaaaa8qacaWFGcGaamyAaiabg2da9iaadMgacqGHRaWkcaaIXaaa aa@3C51@

  • Using initial π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiWda haaa@3841@ to generate a sample  R= r i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGGcGaamOuaiabg2da9iaadkhapaWaaSbaaKqbGeaapeGa amyAaaqcfa4daeqaaaaa@3C95@ ,  i=1,,m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGGcGaamyAaiabg2da9iaaigdacaGGSaGaeyOjGWRaaiil aiaad2gaaaa@3E57@ using binomial distribution, where r 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOCam aaBaaajuaibaGaaGymaaqcfayabaaaaa@3914@ following the binomial (nm, π) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaGGOaGaamOBaiabgkHiTiaad2gacaGGSaGaaiiOaiabec8a WjaacMcaaaa@3E61@ distribution and the variables r i / r 1 , r 2 ,, r i1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGYbWdamaaBaaajuaibaWdbiaadMgaaKqba+aabeaapeGa ai4laiaadkhapaWaaSbaaKqbGeaapeGaaGymaaqcfa4daeqaa8qaca GGSaGaamOCa8aadaWgaaqcfasaa8qacaaIYaaajuaGpaqabaWdbiaa cYcacqGHMacVcaGGSaGaamOCa8aadaWgaaqcfasaa8qacaWGPbGaey OeI0IaaGymaaqcfa4daeqaaaaa@4829@ follow the binomial (nm j=1 i1 r j , π) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaGGOaGaamOBaiabgkHiTiaad2gacqGHsisldaGfWbqabKqb G8aabaacbmWdbiaa=PgacqGH9aqpcaaIXaaapaqaa8qacaWFPbGaey OeI0IaaGymaaqcfa4daeaapeGaeyyeIuoaaiaadkhapaWaaSbaaKqb GeaapeGaamOAaaqcfa4daeqaa8qacaGGSaGaaiiOaiabec8aWjaacM caaaa@4AE4@ distribution for i=2,3,,m1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGPbGaeyypa0JaaGOmaiaacYcacaaIZaGaaiilaiabgAci 8kaacYcacaWGTbGaeyOeI0IaaGymaaaa@404A@ .
  • Set r m ={ nm j=1 m1 r j   if nm j=1 m1 r j >0  0 otherwise  MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGYbWdamaaBaaajuaibaWdbiaad2gaaKqba+aabeaapeGa eyypa0Zaaiqaa8aabaqbaeqabiqaaaqaa8qacaWGUbGaeyOeI0Iaam yBaiabgkHiTmaawahabeqcfaYdaeaapeGaamOAaiabg2da9iaaigda a8aabaWdbiaad2gacqGHsislcaaIXaaajuaGpaqaa8qacqGHris5aa GaamOCa8aadaWgaaqcfasaa8qacaWGQbGaaiiOaaqcfa4daeqaa8qa caGGGcGaamyAaiaadAgacaGGGcGaamOBaiabgkHiTiaad2gacqGHsi sldaGfWbqabKqbG8aabaWdbiaadQgacqGH9aqpcaaIXaaapaqaa8qa caWGTbGaeyOeI0IaaGymaaqcfa4daeaapeGaeyyeIuoaaiaadkhapa WaaSbaaKqbGeaapeGaamOAaaqcfa4daeqaa8qacqGH+aGpcaaIWaGa aiiOaaWdaeaapeGaaGimaiaacckacaWGVbGaamiDaiaadIgacaWGLb GaamOCaiaadEhacaWGPbGaam4CaiaadwgacaGGGcaaaaGaay5Eaaaa aa@6E72@
  • Generate k_i as a binomial random variable with parameters n-k sum-r sum and p=( e e λ T i1   β T i 1 e e λ T i   β T i )/( 1 e e λ T i1   β T i 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGWbGaeyypa0ZaaeWaa8aabaacbmWdbiaa=vgapaWaaWba aeqajuaibaWdbiabgkHiTiaadwgajuaGpaWaaWbaaKqbGeqabaWdbi abeU7aSjaadsfajuaGpaWaaSbaaKqbGeaapeGaamyAaiabgkHiTiaa igdaa8aabeaapeGaeyOeI0IaaiiOaKqbaoaalaaajuaipaqaa8qacq aHYoGya8aabaWdbiaadsfajuaGpaWaaSbaaKqbGeaapeGaamyAaaWd aeqaaaaaaaWdbiabgkHiTiaaigdaaaqcfaOaeyOeI0Iaa8xza8aada ahaaqabKqbGeaapeGaeyOeI0IaamyzaKqba+aadaahaaqcfasabeaa peGaeq4UdWMaamivaKqba+aadaWgaaqcfasaa8qacaWGPbaapaqaba WdbiabgkHiTiaacckajuaGdaWcaaqcfaYdaeaapeGaeqOSdigapaqa a8qacaWGubqcfa4damaaBaaajuaibaWdbiaadMgaa8aabeaaaaaaaa aaaKqba+qacaGLOaGaayzkaaGaai4lamaabmaapaqaa8qacaaIXaGa eyOeI0Iaa8xza8aadaahaaqabKqbGeaapeGaeyOeI0IaamyzaKqba+ aadaahaaqcfasabeaapeGaeq4UdWMaamivaKqba+aadaWgaaqcfasa a8qacaWGPbGaeyOeI0IaaGymaaWdaeqaa8qacqGHsislcaGGGcqcfa 4aaSaaaKqbG8aabaWdbiabek7aIbWdaeaapeGaamivaKqba+aadaWg aaqcfasaa8qacaWGPbaapaqabaaaaaaapeGaeyOeI0IaaGymaaaaaK qbakaawIcacaGLPaaaaaa@78A5@
  • Step 3: Set k sum= k sum+ k i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGRbGaaiiOaiaadohacaWG1bGaamyBaiabg2da9iaaccka caWGRbGaaiiOaiaadohacaWG1bGaamyBaiabgUcaRiaadUgapaWaaS baaKqbGeaapeGaamyAaaqcfa4daeqaaaaa@468A@ and r sum=r sum+ r i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGYbGaaiiOaiaadohacaWG1bGaamyBaiabg2da9iaadkha caGGGcGaam4CaiaadwhacaWGTbGaey4kaSIaamOCa8aadaWgaaqcfa saa8qacaWGPbaajuaGpaqabaaaaa@457B@ .

    Step 4: If i<m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyAai abgYda8iaad2gaaaa@3969@ , go to step 2; otherwise, stop.

    Percentile bootstrap algorithm (Boot-p)

    We can increase information about the population value more than does a point estimate by using a parametric bootstrap interval. We propose to use confidence intervals based on the parameteric bootstrap methods using percentile bootstrap Algorithm (Boot-p) based on the idea of Efron [16].

    The algorithm for estimating the confidence intervals is illustrated as follows:

    Before progressing further, we first describe how we generate progressively interval Type I censored data with binomial random removals. The following algorithm is followed to obtain these samples.

    1. Specify the values of n;m;T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad6gaca GG7aGaamyBaiaacUdacaWGubaaaa@3AB5@ .
    2. Specify the values of λ,β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSj aacYcacqaHYoGyaaa@3A7E@ and π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@ .
    3. Form data algorithm; compute the maximum likelihood estimates of the parameters λ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafq4UdW MbaKaaaaa@3849@ , β ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqOSdi MbaKaaaaa@3836@ and π ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiWda NbaKaaaaa@3852@ , by solving the likelihood equations simultaneously in (8), (9) and (10).
    4. Use λ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafq4UdW MbaKaaaaa@3849@ , β ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqOSdi MbaKaaaaa@3836@ and π ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiWda NbaKaaaaa@3852@ , to generate a bootstrap sample k * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGRbWdamaaCaaabeqaa8qacaGGQaaaaaaa@3884@ with the same values of r_i, m;(i=1,2,…,m) using algorithm presented in Balakrishnan & Sandhu [17].
    5. As in step 3, based on k * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGRbWdamaaCaaabeqaa8qacaGGQaaaaaaa@3884@ compute the bootstrap sample estimates of λ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafq4UdW MbaKaaaaa@3849@ , β ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqOSdi MbaKaaaaa@3836@ and π ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiWda NbaKaaaaa@3852@ , say λ ^ * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafq4UdW MbaKaacaGGQaaaaa@38F7@ , β ^ * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqOSdi MbaKaacaGGQaaaaa@38E4@ and π ^ * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiWda NbaKaacaGGQaaaaa@3900@ .
    6. Repeat steps 4-5 B times representing B bootstrap maximum likelihood estimators of λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSb aa@382D@ , β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIb aa@381A@ and π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@ based on B different bootstrap samples.
    7. Arrange all λ ^ *'s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafq4UdW MbaKaacaGGQaGaai4jaiaadohaaaa@3A9A@ , β ^ *'s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqOSdi MbaKaacaGGQaGaai4jaiaadohaaaa@3A87@ and π ^ *'s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiWda NbaKaacaGGQaGaai4jaiaadohaaaa@3AA3@ , in an ascending order to obtain the bootstrap sample ( φ l [ 1 ] , φ l [ 2 ] ,, φ l [ B ] ), l=1,2,3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaqadaWdaeaapeGaeqOXdO2damaaDaaajuaibaWdbiaadYga a8aabaqcfa4dbmaadmaajuaipaqaa8qacaaIXaaacaGLBbGaayzxaa aaaKqbakaacYcacqaHgpGApaWaa0baaKqbGeaapeGaamiBaaWdaeaa juaGpeWaamWaaKqbG8aabaWdbiaaikdaaiaawUfacaGLDbaaaaqcfa OaaiilaiabgAci8kaacYcacqaHgpGApaWaa0baaKqbGeaapeGaamiB aaWdaeaajuaGpeWaamWaaKqbG8aabaWdbiaadkeaaiaawUfacaGLDb aaaaaajuaGcaGLOaGaayzkaaGaaiilaiaacckacaWGSbGaeyypa0Ja aGymaiaacYcacaaIYaGaaiilaiaaiodaaaa@594D@ (where φ 1 λ ^ * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHgpGApaWaaSbaaKqbGeaapeGaaGymaaqcfa4daeqaa8qa cqGHHjIUcuaH7oaBgaqcaiaacQcaaaa@3E73@ ,   φ 2 β ^ * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaGGGcGaeqOXdO2damaaBaaajuaibaWdbiaaikdaaKqba+aa beaapeGaeyyyIORafqOSdiMbaKaacaGGQaaaaa@3F85@ and φ 3 π ^ * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHgpGApaWaaSbaaKqbGeaapeGaaG4maaqcfa4daeqaa8qa cqGHHjIUcuaHapaCgaqcaiaacQcaaaa@3E7E@ ).

    Let G( z )=P( φ l z ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGhbWaaeWaa8aabaWdbiaadQhaaiaawIcacaGLPaaacqGH 9aqpcaWGqbWaaeWaa8aabaWdbiabeA8aQ9aadaWgaaqcfasaa8qaca WGSbaajuaGpaqabaWdbiabgsMiJkaadQhaaiaawIcacaGLPaaaaaa@4418@ be the cumulative distribution function of φ l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqOXdO2aaSbaaKqbGeaacaWGSbaajuaGbeaaaaa@3A24@ . Define φ lboot = G 1 ( z ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHgpGApaWaaSbaaKqbGeaapeGaamiBaiaadkgacaWGVbGa am4BaiaadshaaKqba+aabeaapeGaeyypa0Jaam4ra8aadaahaaqabK qbGeaapeGaeyOeI0IaaGymaaaajuaGdaqadaWdaeaapeGaamOEaaGa ayjkaiaawMcaaaaa@4554@ for given Z. The approximate bootstrap 100( 12γ )% MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaaIXaGaaGimaiaaicdadaqadaWdaeaapeGaaGymaiabgkHi TiaaikdacqaHZoWzaiaawIcacaGLPaaacaGGLaaaaa@3F30@ confidence interval (ABCI) of φ l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqOXdO2aaSbaaKqbGeaacaWGSbaajuaGbeaaaaa@3A24@ is given by [ φ l boot ( γ ), φ l boot ( 1γ ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaamWaaeaacqaHgpGAdaWgaaqcfasaaiaadYgaaeqaaKqbaoaa BaaajuaibaGaamOyaiaad+gacaWGVbGaamiDaaqabaqcfa4aaeWaae aacqaHZoWzaiaawIcacaGLPaaacaGGSaGaeqOXdO2aaSbaaKqbGeaa caWGSbaabeaajuaGdaWgaaqcfasaaiaadkgacaWGVbGaam4Baiaads haaeqaaKqbaoaabmaabaGaaGymaiabgkHiTiabeo7aNbGaayjkaiaa wMcaaaGaay5waiaaw2faaaaa@51A3@ .

    Bayesian Estimation and MCMC Technique

    In this section, we will focus to Bayesian approach using Markov chain Monte Carlo (MCMC) method to generate from the posterior distributions and in turn computing the Bayes estimators are developed.

    Bayesian estimation

    In Bayesian scenario, we need to assume the prior distribution of the unknown model parameters to take into account uncertainty of the parameters. The informative prior densities for λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSb aa@382D@ and β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIb aa@381A@ are given as

    g 1 ( λ )α  λ b1 e λa ,a,b,λ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadEgada WgaaqcfasaaiaaigdaaKqbagqaamaabmaabaGaeq4UdWgacaGLOaGa ayzkaaGaeqySdegeaaaaaaaaa8qacaGGGcGaeq4UdW2aaWbaaeqaju aibaGaamOyaiabgkHiTiaaigdaaaqcfaOaamyzamaaCaaabeqcfasa aiabgkHiTiabeU7aSjaadggaaaqcfaOaaiilaiaadggacaGGSaGaam OyaiaacYcacqaH7oaBcqGH+aGpcaaIWaaaaa@50E0@ ,

    g 2 ( β ) α  β d1 e βc ,c,d,β > 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadEgada WgaaqcfasaaiaaikdaaKqbagqaamaabmaabaGaeqOSdigacaGLOaGa ayzkaaaeaaaaaaaaa8qacaGGGcWdaiabeg7aH9qacaGGGcWdaiabek 7aInaaCaaabeqcfasaaiaadsgacqGHsislcaaIXaaaaKqbakaadwga daahaaqabKqbGeaacqGHsislcqaHYoGycaWGJbaaaKqbakaacYcaca WGJbGaaiilaiaadsgacaGGSaGaeqOSdi2dbiaacckapaGaeyOpa4Zd biaacckapaGaaGimaaaa@5475@ ,

    and π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@ has a

    g 3 ( π ) α  π A1 ( 1π ) B1 ,0<π<1; A,B>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadEgada WgaaqcfasaaiaaiodaaKqbagqaamaabmaabaGaeqiWdahacaGLOaGa ayzkaaaeaaaaaaaaa8qacaGGGcWdaiabeg7aH9qacaGGGcGaeqiWda 3aaWbaaeqajuaibaGaamyqaiabgkHiTiaaigdaaaqcfa4aaeWaaeaa caaIXaGaeyOeI0IaeqiWdahacaGLOaGaayzkaaWaaWbaaeqajuaiba GaamOqaiabgkHiTiaaigdaaaqcfaOaaiilaiaaicdacqGH8aapcqaH apaCcqGH8aapcaaIXaGaai4oaiaacckacaGGbbGaaiilaiaackeacq GH+aGpcaaIWaaaaa@5979@

    Note that the parameters λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSb aa@382D@ , β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIb aa@381A@ and π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@ behave as independent random variables. The joint informative prior probability density function of λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSb aa@382D@ , β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIb aa@381A@ and π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@ is

    g( λ,β,π )α  g 1 ( λ )× g 2 ( β )× g 3 ( π ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadEgada qadaqaaiabeU7aSjaacYcacqaHYoGycaGGSaGaeqiWdahacaGLOaGa ayzkaaGaeqySdegeaaaaaaaaa8qacaGGGcWdaiaadEgadaWgaaqcfa saaiaaigdaaKqbagqaamaabmaabaGaeq4UdWgacaGLOaGaayzkaaGa ey41aqRaam4zamaaBaaajuaibaGaaGOmaaqcfayabaWaaeWaaeaacq aHYoGyaiaawIcacaGLPaaacqGHxdaTcaWGNbWaaSbaaKqbGeaacaaI ZaaajuaGbeaadaqadaqaaiabec8aWbGaayjkaiaawMcaaaaa@57BC@

    ( λ,β,π ) α  λ b1 e λa β d1 e βc π A1 ( 1π ) B1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaeWaaeaacqaH7oaBcaGGSaGaeqOSdiMaaiilaiabec8aWbGa ayjkaiaawMcaaiaacckacqaHXoqycaGGGcGaeq4UdW2aaWbaaKqbGe qabaGaamOyaiabgkHiTiaaigdaaaqcfaOaamyzamaaCaaajuaibeqa aiabgkHiTiabeU7aSjaadggaaaqcfaOaeqOSdi2aaWbaaKqbGeqaba GaamizaiabgkHiTiaaigdaaaqcfaOaamyzamaaCaaabeqcfasaaiab gkHiTiabek7aIjaadogaaaqcfaOaeqiWda3aaWbaaeqajuaibaGaam yqaiabgkHiTiaaigdaaaqcfa4aaeWaaeaacaaIXaGaeyOeI0IaeqiW dahacaGLOaGaayzkaaWaaWbaaeqajuaibaGaamOqaiabgkHiTiaaig daaaaaaa@63EF@ (11)

    where a,b,c,d,A and B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadggaca GGSaGaamOyaiaacYcacaWGJbGaaiilaiaadsgacaGGSaGaamyqaaba aaaaaaaapeGaaiiOaiaadggacaWGUbGaamizaiaacckacaWGcbaaaa@438E@ are assumed to be known and are chosen to reflect prior knowledge about λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSb aa@382D@ , β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIb aa@381A@ and π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@ . Note that when a=b=c=d=A=B=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadggacq GH9aqpcaWGIbGaeyypa0Jaam4yaiabg2da9iaadsgacqGH9aqpcaWG bbGaeyypa0JaamOqaiabg2da9iaaicdaaaa@4282@ , (we call it prior 0) they are the non-informative λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSb aa@382D@ , β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIb aa@381A@ and π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@ respectively.

    It follows from (4), (6) and (11) that the joint posterior density function of λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSb aa@382D@ , β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIb aa@381A@ and π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@ given x is thus

    π*( λ,β,π )α L 1 ( X;λ,β/R=r )×P( R )×g( λ,β,π ) 0 0 0 1 L 1 ( X;λ,β/R=r )×P( R )g( λ,β,π )dλdβdπ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWj aacQcadaqadaqaaiabeU7aSjaacYcacqaHYoGycaGGSaGaeqiWdaha caGLOaGaayzkaaGaeqySde2aaSaaaeaacaWGmbWaaSbaaKqbGeaaca aIXaaajuaGbeaadaqadaqaaiaadIfacaGG7aGaeq4UdWMaaiilaiab ek7aIjaac+cacaWGsbGaeyypa0JaamOCaaGaayjkaiaawMcaaiabgE na0kaadcfadaqadaqaaiaadkfaaiaawIcacaGLPaaacqGHxdaTcaWG NbWaaeWaaeaacqaH7oaBcaGGSaGaeqOSdiMaaiilaiabec8aWbGaay jkaiaawMcaaaqaamaapedabaWaa8qmaeaadaWdXaqaaiaadYeadaWg aaqcfasaaiaaigdaaKqbagqaamaabmaabaGaamiwaiaacUdacqaH7o aBcaGGSaGaeqOSdiMaai4laiaackfacqGH9aqpcaGGYbaacaGLOaGa ayzkaaGaey41aqRaamiuamaabmaabaGaamOuaaGaayjkaiaawMcaai aadEgadaqadaqaaiabeU7aSjaacYcacqaHYoGycaGGSaGaeqiWdaha caGLOaGaayzkaaGaamizaiabeU7aSjaadsgacqaHYoGycaWGKbGaeq iWdahajuaibaGaaGimaaqaaiaaigdaaKqbakabgUIiYdaajuaibaGa aGimaaqaaiabg6HiLcqcfaOaey4kIipaaKqbGeaacaaIWaaabaGaey OhIukajuaGcqGHRiI8aaaaaaa@913C@

    π*( λ,β,π )α i=1 m [ 1 e u( T i ) e u( T i1 ) ] k i e { k i u( T i1 )+ r i u( T i ) } π j=1 m1 r j . ( 1π ) ( m1 )( nm ) j=1 m1 ( mj ) r j  ×  λ b1 e λa β d1 e βc π A1 ( 1π ) B1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWj aacQcadaqadaqaaiabeU7aSjaacYcacqaHYoGycaGGSaGaeqiWdaha caGLOaGaayzkaaGaeqySde2aaCbmaeaacqGHpis1aKqbGeaacaWGPb Gaeyypa0JaaGymaaqaaiaad2gaaaqcfa4aamWaaeaacaaIXaGaeyOe I0YaaSaaaeaacaWGLbWaaWbaaeqajuaibaGaeyOeI0IaamyDaKqbao aabmaajuaibaGaamivaKqbaoaaBaaajuaibaGaamyAaaqabaaacaGL OaGaayzkaaaaaaqcfayaaiaadwgadaahaaqabKqbGeaacqGHsislca WG1bqcfa4aaeWaaKqbGeaacaWGubqcfa4aaSbaaKqbGeaacaWGPbGa eyOeI0IaaGymaaqabaaacaGLOaGaayzkaaaaaaaaaKqbakaawUfaca GLDbaadaahaaqabKqbGeaacaWGRbqcfa4aaSbaaKqbGeaacaWGPbaa beaaaaqcfaOaamyzamaaCaaajuaibeqaaiabgkHiTKqbaoaacmaaju aibaGaam4AaKqbaoaaBaaajuaibaGaamyAaaqabaGaamyDaKqbaoaa bmaajuaibaGaamivaKqbaoaaBaaajuaibaGaamyAaiabgkHiTiaaig daaeqaaaGaayjkaiaawMcaaiabgUcaRiaadkhajuaGdaWgaaqcfasa aiaadMgaaeqaaiaadwhajuaGdaqadaqcfasaaiaadsfajuaGdaWgaa qcfasaaiaadMgaaeqaaaGaayjkaiaawMcaaaGaay5Eaiaaw2haaaaa cqaHapaCjuaGdaaeWaqcfasaaiaadkhajuaGdaWgaaqcfasaaiaadQ gaaeqaaaqaaiaadQgacqGH9aqpcaaIXaaabaGaamyBaiabgkHiTiaa igdaaiabggHiLdqcfaOaaiOlamaabmaabaGaaGymaiabgkHiTiabec 8aWbGaayjkaiaawMcaamaaCaaajuaibeqaaKqbaoaabmaajuaibaGa amyBaiabgkHiTiaaigdaaiaawIcacaGLPaaajuaGdaqadaqcfasaai aad6gacqGHsislcaWGTbaacaGLOaGaayzkaaGaeyOeI0scfa4aaabm aKqbGeaajuaGdaqadaqcfasaaiaad2gacqGHsislcaWGQbaacaGLOa GaayzkaaGaamOCaKqbaoaaBaaajuaibaGaamOAaaqabaaabaGaamOA aiabg2da9iaaigdaaeaacaWGTbGaeyOeI0IaaGymaaGaeyyeIuoaaa qcfaieaaaaaaaaa8qacaGGGcWdaiabgEna0+qacaGGGcWdaiabeU7a SnaaCaaabeqcfasaaiaadkgacqGHsislcaaIXaaaaKqbakaadwgada ahaaqcfasabeaacqGHsislcqaH7oaBcaWGHbaaaKqbakabek7aInaa CaaajuaibeqaaiaadsgacqGHsislcaaIXaaaaKqbakaadwgadaahaa qabKqbGeaacqGHsislcqaHYoGycaWGJbaaaKqbakabec8aWnaaCaaa juaibeqaaiaadgeacqGHsislcaaIXaaaaKqbaoaabmaabaGaaGymai abgkHiTiabec8aWbGaayjkaiaawMcaamaaCaaabeqcfasaaiaadkea cqGHsislcaaIXaaaaaaa@CFFC@ (12)

    where u( T i )= e λ T i β T i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaajwhada qadaqaaiaadsfadaWgaaqcfasaaiaadMgaaKqbagqaaaGaayjkaiaa wMcaaiabg2da9iaadwgadaahaaqabKqbGeaacqaH7oaBcaWGubqcfa 4aaSbaaKqbGeaacaWGPbaabeaacqGHsisljuaGdaWcaaqcfasaaiab ek7aIbqaaiaadsfajuaGdaWgaaqcfasaaiaadMgaaeqaaaaaaaaaaa@483C@ and u( T i1 )= e λ T i1 β T i1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaajwhada qadaqaaiaadsfadaWgaaqcfasaaiaadMgacqGHsislcaaIXaaabeaa aKqbakaawIcacaGLPaaacqGH9aqpcaWGLbWaaWbaaKqbGeqabaGaeq 4UdWMaamivaKqbaoaaBaaajuaibaGaamyAaiabgkHiTiaaigdaaeqa aiabgkHiTKqbaoaalaaajuaibaGaeqOSdigabaGaamivaKqbaoaaBa aajuaibaGaamyAaiabgkHiTiaaigdaaeqaaaaaaaaaaa@4D34@ .

    It is not possible to compute (12) analytically. The problem is that the integrals in (12) are usually impossible to evaluate analytically, and the numerical methods may fail. The MCMC method provides an alternative method for parameter estimation. In the following subsections, we propose using the MCMC technique to obtain Bayes estimates of the unknown parameters and construct the corresponding credible intervals.

    MCMC technique

    Computer simulation of Markov chains in the space of parameter will depend on Markov chain Monte Carlo (MCMC) Gilks et al. [18]. The Markov chains are defined in such a way that the posterior distribution in the given statistical inference problem is the asymptotic distribution. However, the posterior likelihood usually does not have a closed form for a given progressively type-I interval-censored data. Moreover, a numerical integration cannot be easily applied in this situation. A lot of standard approaches to display like Markov chains exist, including Gibbs sampling, Metropolis-Hastings (M-H) and reversible jump. The M-H algorithm is a very general MCMC method first expansion by Metropolis et al. [19] and later extended by Hastings [20]. it is possible to use these algorithms by implement posterior simulation in essentially any problem which allow point wise evaluation of the prior distribution and likelihood function. It can be used to obtain random samples from any arbitrarily complicated target distribution of any dimension that is known up to a normalizing constant. In fact, Gibbs sampler is just a special case of the M-H algorithm.

    In order to use the method of MCMC for estimating the parameters of the flexible Weibull distribution and random removal, namely, λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSb aa@382D@ , β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIb aa@381A@ and π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@ . Let us consider independent priors as in (10), the full conditional distribution for any parameter can be obtained, to within a constant, by factoring out from the likelihood function L(X,R;λ,β) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYeaca GGOaGaamiwaiaacYcacaWGsbGaai4oaiabeU7aSjaacYcacqaHYoGy caGGPaaaaa@3FCB@ any terms containing the relevant parameter and multiplying by its prior. From (11), the full posterior conditional distribution for λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSb aa@382D@ is proportional to

    π * ( λ/x,β,π ) i=1 m [ 1 u( T i ) u( T i1 ) ] k i e { k i u( T i1 )+  r i u( T i ) } × λ b1 e λa MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHapaCpaWaaWbaaeqabaWdbiaacQcaaaWaaeWaa8aabaWd biabeU7aSjaac+cacaWG4bGaaiilaiabek7aIjaacYcacqaHapaCai aawIcacaGLPaaacqGHDisTdaGfWbqabKqbG8aabaWdbiaadMgacqGH 9aqpcaaIXaaapaqaa8qacaWGTbaajuaGpaqaa8qacqGHpis1aaWaam Waa8aabaWdbiaaigdacqGHsisldaWcaaWdaeaapeGaamyDamaabmaa paqaa8qacaWGubWdamaaBaaajuaibaWdbiaadMgaaKqba+aabeaaa8 qacaGLOaGaayzkaaaapaqaa8qacaWG1bWaaeWaa8aabaWdbiaadsfa paWaaSbaaKqbGeaapeGaamyAaiabgkHiTiaaigdaaKqba+aabeaaa8 qacaGLOaGaayzkaaaaaaGaay5waiaaw2faa8aadaahaaqabeaapeGa am4Aa8aadaWgaaqcfasaa8qacaWGPbaajuaGpaqabaaaa8qacaWGLb WdamaaCaaabeqcfasaa8qacqGHsisljuaGdaGadaqcfaYdaeaapeGa am4AaKqba+aadaWgaaqcfasaa8qacaWGPbaapaqabaWdbiaadwhaju aGdaqadaqcfaYdaeaapeGaamivaKqba+aadaWgaaqcfasaa8qacaWG PbGaeyOeI0IaaGymaaWdaeqaaaWdbiaawIcacaGLPaaacqGHRaWkca GGGcGaamOCaKqba+aadaWgaaqcfasaa8qacaWGPbaapaqabaWdbiaa dwhajuaGdaqadaqcfaYdaeaapeGaamivaKqba+aadaWgaaqcfasaa8 qacaWGPbaapaqabaaapeGaayjkaiaawMcaaaGaay5Eaiaaw2haaaaa juaGcqGHxdaTcqaH7oaBpaWaaWbaaeqajuaibaWdbiaadkgacqGHsi slcaaIXaaaaKqbakaadwgapaWaaWbaaeqajuaibaWdbiabgkHiTiab eU7aSjaadggaaaaaaa@87A4@ (12)

    Also, the full posterior conditional distribution for β is proportional to

    π * ( β/x,λ,π ) i=1 m [ 1 u( T i ) u( T i1 ) ] k i e { k i u( T i1 )+  r i u( T i ) } × β d1 e βc MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHapaCpaWaaWbaaeqabaWdbiaacQcaaaWaaeWaa8aabaWd biabek7aIjaac+cacaWG4bGaaiilaiabeU7aSjaacYcacqaHapaCai aawIcacaGLPaaacqGHDisTdaGfWbqabKqbG8aabaWdbiaadMgacqGH 9aqpcaaIXaaapaqaa8qacaWGTbaajuaGpaqaa8qacqGHpis1aaWaam Waa8aabaWdbiaaigdacqGHsisldaWcaaWdaeaapeGaamyDamaabmaa paqaa8qacaWGubWdamaaBaaajuaibaWdbiaadMgaaKqba+aabeaaa8 qacaGLOaGaayzkaaaapaqaa8qacaWG1bWaaeWaa8aabaWdbiaadsfa paWaaSbaaKqbGeaapeGaamyAaiabgkHiTiaaigdaaKqba+aabeaaa8 qacaGLOaGaayzkaaaaaaGaay5waiaaw2faa8aadaahaaqabeaapeGa am4Aa8aadaWgaaqcfasaa8qacaWGPbaajuaGpaqabaaaa8qacaWGLb WdamaaCaaabeqcfasaa8qacqGHsisljuaGdaGadaqcfaYdaeaapeGa am4AaKqba+aadaWgaaqcfasaa8qacaWGPbaapaqabaWdbiaadwhaju aGdaqadaqcfaYdaeaapeGaamivaKqba+aadaWgaaqcfasaa8qacaWG PbGaeyOeI0IaaGymaaWdaeqaaaWdbiaawIcacaGLPaaacqGHRaWkca GGGcGaamOCaKqba+aadaWgaaqcfasaa8qacaWGPbaapaqabaWdbiaa dwhajuaGdaqadaqcfaYdaeaapeGaamivaKqba+aadaWgaaqcfasaa8 qacaWGPbaapaqabaaapeGaayjkaiaawMcaaaGaay5Eaiaaw2haaaaa juaGcqGHxdaTcqaHYoGypaWaaWbaaeqajuaibaWdbiaadsgacqGHsi slcaaIXaaaaKqbakaadwgapaWaaWbaaeqajuaibaWdbiabgkHiTiab ek7aIjaadogaaaaaaa@8782@ (13)

    Similarly, the marginal posterior density of π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@ is proportional to

    π * ( π/x,λ,β ) π A+ j=1 m1 r j 1 ( 1π ) B+( m1 )( nm ) j=1 m1 ( mj ) r j 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHapaCpaWaaWbaaeqabaWdbiaacQcaaaWaaeWaa8aabaWd biabec8aWjaac+cacaWG4bGaaiilaiabeU7aSjaacYcacqaHYoGyai aawIcacaGLPaaacqGHDisTcqaHapaCpaWaaWbaaeqajuaibaWdbiaa dgeacqGHRaWkjuaGdaGfWbqcfasab8aabaWdbiaadQgacqGH9aqpca aIXaaapaqaa8qacaWGTbGaeyOeI0IaaGymaaWdaeaapeGaeyyeIuoa aiaadkhajuaGpaWaaSbaaKqbGeaapeGaamOAaaWdaeqaa8qacqGHsi slcaaIXaaaaKqbaoaabmaapaqaa8qacaaIXaGaeyOeI0IaeqiWdaha caGLOaGaayzkaaWdamaaCaaabeqcfasaa8qacaWGcbGaey4kaSscfa 4aaeWaaKqbG8aabaWdbiaad2gacqGHsislcaaIXaaacaGLOaGaayzk aaqcfa4aaeWaaKqbG8aabaWdbiaad6gacqGHsislcaWGTbaacaGLOa GaayzkaaGaeyOeI0scfa4aaybCaKqbGeqapaqaa8qacaWGQbGaeyyp a0JaaGymaaWdaeaapeGaamyBaiabgkHiTiaaigdaa8aabaWdbiabgg HiLdaajuaGdaqadaqcfaYdaeaapeGaamyBaiabgkHiTiaadQgaaiaa wIcacaGLPaaacaWGYbqcfa4damaaBaaajuaibaWdbiaadQgaa8aabe aapeGaeyOeI0IaaGymaaaaaaa@7AF0@ (14)

    It is noted that the posterior distribution of π MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHapaCaaa@3862@ is beta with parameters A* MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeaca GGQaaaaa@37ED@ and B* MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkeaca GGQaaaaa@37EE@ where A * =A+ j=1 m1 r j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGbbWdamaaCaaabeqaa8qacaGGQaaaaiabg2da9iaabgea cqGHRaWkdaGfWbqabKqbG8aabaWdbiaabQgacqGH9aqpcaaIXaaapa qaa8qacaqGTbGaeyOeI0IaaGymaaqcfa4daeaapeGaeyyeIuoaaiaa bkhapaWaaSbaaKqbGeaapeGaaeOAaaqcfa4daeqaaaaa@4684@ and B * =B+( m1 )( nm ) j=1 m1 ( mj ) r j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGcbWdamaaCaaabeqaa8qacaGGQaaaaiabg2da9iaabkea cqGHRaWkdaqadaWdaeaapeGaaeyBaiabgkHiTiaaigdaaiaawIcaca GLPaaadaqadaWdaeaapeGaaeOBaiabgkHiTiaab2gaaiaawIcacaGL PaaacqGHsisldaGfWbqabKqbG8aabaWdbiaabQgacqGH9aqpcaaIXa aapaqaa8qacaqGTbGaeyOeI0IaaGymaaqcfa4daeaapeGaeyyeIuoa amaabmaapaqaa8qacaqGTbGaeyOeI0IaaeOAaaGaayjkaiaawMcaai aabkhapaWaaSbaaKqbGeaapeGaaeOAaaqcfa4daeqaaaaa@549B@ and, π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@ therefore, samples of can be easily generated using any beta generating routine. But the conditional posterior distribution of λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSb aa@382D@ and β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIb aa@381A@ equations (12) and (13) respectively, cannot be reduced analytically to well-known distributions and therefore it is not possible to sample directly by standard methods, but the plot of it show that it is similar to normal distribution. So to generate random numbers from this distribution, we use the M-H method with normal proposal distribution.

    MCMC process

    Now, we propose the following scheme to generate λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSb aa@382D@ , β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIb aa@381A@ and π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@ from density functions and in turn obtain the Bayes estimates and the corresponding credible intervals.

    1. Start with an λ ( 0 ) = λ ^ , β ( 0 ) = β ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSn aaCaaabeqcfasaaKqbaoaabmaajuaibaGaaGimaaGaayjkaiaawMca aaaajuaGcqGH9aqpcuaH7oaBgaqcaiaacYcacqaHYoGydaahaaqabK qbGeaajuaGdaqadaqcfasaaiaaicdaaiaawIcacaGLPaaaaaqcfaOa eyypa0JafqOSdiMbaKaaaaa@47B9@ and M=burnin MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad2eacq GH9aqpcaWGIbGaamyDaiaadkhacaWGUbGaeyOeI0IaamyAaiaad6ga aaa@3EEA@ .
    2. Set t=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadshacq GH9aqpcaaIXaaaaa@3933@ .
    3. Generate π ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiWda 3aaWbaaeqajuaibaqcfa4aaeWaaKqbGeaacaWG0baacaGLOaGaayzk aaaaaaaa@3BD0@ from beta distribution π*( π/x,λ,β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWj aacQcadaqadaqaaiabec8aWjaac+cacaWG4bGaaiilaiabeU7aSjaa cYcacqaHYoGyaiaawIcacaGLPaaaaaa@428F@ .
    4. Using M-H algorithm Metropolis et al. [19], λ ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSn aaCaaabeqcfasaaKqbaoaabmaajuaibaGaamiDaaGaayjkaiaawMca aaaaaaa@3BBB@ from π*( λ/x,β,π ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWj aacQcadaqadaqaaiabeU7aSjaac+cacaWG4bGaaiilaiabek7aIjaa cYcacqaHapaCaiaawIcacaGLPaaaaaa@428F@ with the N( λ ( t1 ) , σ λ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad6eada qadaqaaiabeU7aSnaaCaaabeqcfasaaKqbaoaabmaajuaibaGaamiD aiabgkHiTiaaigdaaiaawIcacaGLPaaaaaqcfaOaaiilaiabeo8aZn aaDaaajuaibaGaeq4UdWgabaGaaGOmaaaaaKqbakaawIcacaGLPaaa aaa@460E@ proposal distribution where σ λ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo8aZn aaDaaajuaibaGaeq4UdWgabaGaaGOmaaaaaaa@3AFC@ is the variance of obtained using variance-covariance matrix; similarly, β ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIn aaCaaabeqcfasaaKqbaoaabmaajuaibaGaamiDaaGaayjkaiaawMca aaaaaaa@3BA8@ from π*( β/X,λ,π ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWj aacQcadaqadaqaaiabek7aIjaac+cacaWGybGaaiilaiabeU7aSjaa cYcacqaHapaCaiaawIcacaGLPaaaaaa@426F@ with the N( β ( t1 ) , σ β 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad6eada qadaqaaiabek7aInaaCaaabeqcfasaaKqbaoaabmaajuaibaGaamiD aiabgkHiTiaaigdaaiaawIcacaGLPaaaaaqcfaOaaiilaiabeo8aZn aaDaaajuaibaGaeqOSdigabaGaaGOmaaaaaKqbakaawIcacaGLPaaa aaa@45E8@ proposal distribution where σ β 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo8aZn aaDaaajuaibaGaeqOSdigabaGaaGOmaaaaaaa@3AE9@ is the variance of β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIb aa@381A@ obtained using variance-covariance matrix.
    5. Compute λ ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSn aaCaaabeqcfasaaKqbaoaabmaajuaibaGaamiDaaGaayjkaiaawMca aaaaaaa@3BBB@ , λ ( 0 ) = λ ^ , β ( 0 ) = β ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSn aaCaaabeqcfasaaKqbaoaabmaajuaibaGaaGimaaGaayjkaiaawMca aaaajuaGcqGH9aqpcuaH7oaBgaqcaiaacYcacqaHYoGydaahaaqabK qbGeaajuaGdaqadaqcfasaaiaaicdaaiaawIcacaGLPaaaaaqcfaOa eyypa0JafqOSdiMbaKaaaaa@47B9@ and π ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWn aaCaaabeqcfasaaKqbaoaabmaajuaibaGaamiDaaGaayjkaiaawMca aaaaaaa@3BC4@ .
    6. Set t=t+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadshacq GH9aqpcaWG0bGaey4kaSIaaGymaaaa@3B0E@ .
    7. Repeats Steps 3-6 N times.
    8. Obtain the Bayes estimates of λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSb aa@382D@ , β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIb aa@381A@ and π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@ with respect to the squared error loss function as E ^ ( λ/x )= 1 NM i=M+1 N λ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadweaga qcamaabmaabaGaeq4UdWMaai4laiaadIhaaiaawIcacaGLPaaacqGH 9aqpdaWcaaqaaiaaigdaaeaacaWGobGaeyOeI0IaamytaaaadaaeWb qaaiabeU7aSnaaBaaajuaibaGaamyAaaqcfayabaaajuaibaGaamyA aiabg2da9iaad2eacqGHRaWkcaaIXaaabaGaamOtaaqcfaOaeyyeIu oaaaa@4C3E@ , E ^ ( β/x )= 1 NM i=M+1 N β i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadweaga qcamaabmaabaGaeqOSdiMaai4laiaadIhaaiaawIcacaGLPaaacqGH 9aqpdaWcaaqaaiaaigdaaeaacaWGobGaeyOeI0IaamytaaaadaaeWb qaaiabek7aInaaBaaajuaibaGaamyAaaqcfayabaaajuaibaGaamyA aiabg2da9iaad2eacqGHRaWkcaaIXaaabaGaamOtaaqcfaOaeyyeIu oaaaa@4C18@ and E ^ ( π/x )= 1 NM i=M+1 N π i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadweaga qcamaabmaabaGaeqiWdaNaai4laiaadIhaaiaawIcacaGLPaaacqGH 9aqpdaWcaaqaaiaaigdaaeaacaWGobGaeyOeI0IaamytaaaadaaeWb qaaiabec8aWnaaBaaajuaibaGaamyAaaqcfayabaaajuaibaGaamyA aiabg2da9iaad2eacqGHRaWkcaaIXaaabaGaamOtaaqcfaOaeyyeIu oaaaa@4C50@
    9. To compute the credible intervals of , and , order λ 1 ,..., λ NM MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSn aaBaaajuaibaGaaGymaaqcfayabaGaaiilaiaac6cacaGGUaGaaiOl aiaacYcacqaH7oaBdaWgaaqaaKqbGiaad6eacqGHsislcaWGnbaaju aGbeaaaaa@425E@ , β 1 ,..., β NM MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIn aaBaaajuaibaGaaGymaaqcfayabaGaaiilaiaac6cacaGGUaGaaiOl aiaacYcacqaHYoGydaWgaaqaaKqbGiaad6eacqGHsislcaWGnbaaju aGbeaaaaa@4238@ and π 1 ,..., π NM MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWn aaBaaajuaibaGaaGymaaqcfayabaGaaiilaiaac6cacaGGUaGaaiOl aiaacYcacqaHapaCdaWgaaqaaKqbGiaad6eacqGHsislcaWGnbaaju aGbeaaaaa@4270@ as λ 1 <...< λ NM, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSn aaBaaajuaibaGaaGymaaqcfayabaGaeyipaWJaaiOlaiaac6cacaGG UaGaeyipaWJaeq4UdW2aaSbaaeaajuaicaWGobGaeyOeI0IaamytaK qbakaacYcaaeqaaaaa@43B6@ , β 1 <...< β NM, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIn aaBaaajuaibaGaaGymaaqcfayabaGaeyipaWJaaiOlaiaac6cacaGG UaGaeyipaWJaeqOSdi2aaSbaaeaajuaicaWGobGaeyOeI0IaamytaK qbakaacYcaaeqaaaaa@4390@ and π 1 <...< π NM. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWn aaBaaajuaibaGaaGymaaqcfayabaGaeyipaWJaaiOlaiaac6cacaGG UaGaeyipaWJaeqiWda3aaSbaaeaajuaicaWGobGaeyOeI0Iaamytai aac6caaKqbagqaaaaa@43CA@ .Then the 100( 1γ )% MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaGymai aaicdacaaIWaWaaeWaaeaacaaIXaGaeyOeI0Iaeq4SdCgacaGLOaGa ayzkaaGaaiyjaaaa@3E35@ symmetric credible intervals (SCI) of , and become:

    [ λ ( NM ) γ/2, λ ( NM )( 1γ/2 ) ],[ β ( NM ) γ/2, β ( NM )( 1γ/2 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaaba Gaeq4UdW2aaSbaaKqbGeaajuaGdaqadaqcfasaaiaad6eacqGHsisl caWGnbaacaGLOaGaayzkaaaabeaacqaHZoWzcaGGVaGaaGOmaKqbak aacYcacqaH7oaBdaWgaaqcfasaaKqbaoaabmaajuaibaGaamOtaiab gkHiTiaad2eaaiaawIcacaGLPaaajuaGdaqadaqcfasaaiaaigdacq GHsislcqaHZoWzcaGGVaGaaGOmaaGaayjkaiaawMcaaaqcfayabaaa caGLBbGaayzxaaGaaiilamaadmaabaGaeqOSdi2aaSbaaKqbGeaaju aGdaqadaqcfasaaiaad6eacqGHsislcaWGnbaacaGLOaGaayzkaaaa beaacqaHZoWzcaGGVaGaaGOmaKqbakaacYcacqaHYoGydaWgaaqcfa saaKqbaoaabmaajuaibaGaamOtaiabgkHiTiaad2eaaiaawIcacaGL PaaajuaGdaqadaqcfasaaiaaigdacqGHsislcqaHZoWzcaGGVaGaaG OmaaGaayjkaiaawMcaaaqcfayabaaacaGLBbGaayzxaaaaaa@6E19@ and [ π ( NM )γ/2, π ( NM )( 1γ/2 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaaba GaeqiWda3aaSbaaeaadaqadaqcKvaq=haacaWGobGaeyOeI0Iaamyt aaqcfaIaayjkaiaawMcaaiabeo7aNjaac+cacaaIYaqcfaOaaiilaa qabaGaeqiWda3aaSbaaKqbGeaajuaGdaqadaqcKvaq=haacaWGobGa eyOeI0IaamytaaqcfaIaayjkaiaawMcaaKqbaoaabmaajuaibaGaaG ymaiabgkHiTiabeo7aNjaac+cacaaIYaaacaGLOaGaayzkaaaajuaG beaaaiaawUfacaGLDbaaaaa@54BC@

    Real Data Analysis

    To conduct a study within the Institute of Oncology in Tanta - Egypt. This study is concerned with the treatment of cancerous tumors in blood and studies their impact on the overall health of the patient. Underwent the study 228 patients and they had varying degrees of disease. Patients were examined every 15 days for 6 consecutive months. Of course there were cases of withdrawal (death - interruption of treatment for different reasons)

    As we know on the basis of a single sample, one cannot make a general statement regarding the behavior of proposed estimators, therefore we present a simulation study for the study of the behavior of the estimators in the next section.

    Simulation

    The simulation is conducted using the R version 3.2.2 (for more information about R programming, the reader may refer to this manual of R, version 3.3.0 under development (2015-10-30) Copyright 2000 –2015 R Core Team). The simulation setup is parallel to the real data given in (Table 1). To be specific, each replication of the simulation generates a progressively type-I interval-censored data within twelve subintervals which have pre-specified inspection times (in terms of half month),

    T 0 =0, T 1 =16, T 2 =31, T 3 =46, T 4 =61, T 5 =76, T 6 =91, T 7 =106, T 8 =121, T 9 =136, T 10 =151, T 11 =166 and  T 12 =181. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsfada WgaaqcfasaaiaaicdaaKqbagqaaiabg2da9iaaicdacaGGSaGaaiiv amaaBaaajuaibaGaaGymaaqcfayabaGaeyypa0JaaGymaiaaiAdaca GGSaGaamivamaaBaaajuaibaGaaGOmaaqcfayabaGaeyypa0JaaG4m aiaaigdacaGGSaGaamivamaaBaaajuaibaGaaG4maaqcfayabaGaey ypa0JaaGinaiaaiAdacaGGSaGaaiivamaaBaaajuaibaGaaGinaaqc fayabaGaeyypa0JaaGOnaiaaigdacaGGSaGaamivamaaBaaajuaiba GaaGynaaqcfayabaGaeyypa0JaaG4naiaaiAdacaGGSaGaaiivamaa BaaajuaibaGaaGOnaaqcfayabaGaeyypa0JaaGyoaiaaigdacaGGSa GaaiivamaaBaaajuaibaGaaG4naaqcfayabaGaeyypa0JaaGymaiaa icdacaaI2aGaaiilaiaadsfadaWgaaqcfasaaiaaiIdaaKqbagqaai abg2da9iaaigdacaaIYaGaaGymaiaacYcacaWGubWaaSbaaKqbGeaa caaI5aaajuaGbeaacqGH9aqpcaaIXaGaaG4maiaaiAdacaGGSaGaai ivamaaBaaajuaibaGaaGymaiaaicdaaKqbagqaaiabg2da9iaaigda caaI1aGaaGymaiaacYcacaWGubWaaSbaaKqbGeaacaaIXaGaaGymaa qcfayabaGaeyypa0JaaGymaiaaiAdacaaI2aGcqaaaaaaaaaWdbiaa cckajuaGpaGaamyyaiaad6gacaWGKbGcpeGaaiiOaKqba+aacaWGub WaaSbaaKqbGeaacaaIXaGaaGOmaaqcfayabaGaeyypa0JaaGymaiaa iIdacaaIXaGaaiOlaaaa@8B89@

    The last inspection time, , is the scheduled time to terminate the experiment. The lifetime distribution is flexible Weibull with parameters λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSb aa@382D@ and β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIb aa@381A@ where the simulation input parameters are selected close to the maximum likelihood estimators of flexible Weibull parameters for modeling the real data in (Table 1). The performance of parameter estimation under progressively type I interval censored with random removal is compared via the maximum likelihood, bootstrap method and MCMC procedure developed in this paper. The summary for 1000 simulation runs is shown in (Tables 2-5). Bayes estimates of λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSb aa@382D@ , β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIb aa@381A@ and π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@ using MCMC method, we assume that informative priors a = 2,b = 3,c = 4 , d = 2, A = 2 and B = 3) on λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSb aa@382D@ , β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIb aa@381A@ and π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@ in (Table 4). Also, by non-informative prior using MCMC procedure with Bayes estimation will be obtained on estimates of parameters in (Table 5).

    k i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Aam aaBaaajuaibaGaamyAaaqcfayabaaaaa@393F@

    Cases of Withdrawal

    Number of Random Removals m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyBaa aa@3776@

    Interval in Hours T i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaBaaajuaibaGaamyAaaqcfayabaaaaa@3928@

    Number at Risk

    Number of Failure k i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Aam aaBaaajuaibaGaamyAaaqcfayabaaaaa@393F@

    1

    [0,16)

    228

    25

    2

    2

    [16,31)

    201

    39

    2

    3

    [31,46)

    160

    25

    1

    4

    [46,61)

    134

    20

    3

    5

    [61,76)

    111

    11

    1

    6

    [76,91)

    99

    14

    2

    7

    [91,106)

    83

    11

    3

    8

    [106,121)

    69

    17

    0

    9

    [121,136)

    52

    6

    2

    10

    [136,151)

    44

    31

    1

    11

    [151,166)

    12

    6

    1

    12

    [166,181)

    5

    5

    0

    Table 1: Examine patients every 15 days.

    Different Parameters

    λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW gaaa@3838@

    β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@

    π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiWda haaa@3841@

    Average

    6.441

    0.0841

    0.6312

    MSE

    0.0134

    0.0743

    0.0484

    Bias

    0.0231

    0.1073

    0.0094

    Variance

    0.0129

    0.0627

    0.0483

    ACI

    [5.0132,7.9801]

    [-0.1736,0.0901]

    [0.4421,0.7782]

    Length ACI

    2.9669

    0.2637

    0.3361

    Table 2: Progressively type I interval censored with random removal via the, maximum likelihood.

    Different Parameters

    λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW gaaa@3838@

    β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@

    π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiWda haaa@3841@

    Average

    5.966

    0.099

    0.7058

    MSE

    0.1174

    0.0984

    0.0487

    Bias

    0.0346

    0.1764

    0.0109

    Variance

    0.1162

    0.0673

    0.0486

    ABCI

    [5.0117,8.0412]

    [-0.1811,0.0884]

    [0.4434,0.7992]

    Length ABCI

    3.0295

    0.2695

    0.3558

    Table 3: Progressively type I interval censored with random removal via the, bootstrap method.

    Different Parameters

    λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW gaaa@3838@

    β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@

    π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiWda haaa@3841@

    Average

    4.902

    0.0083

    0.5118

    MSE

    0.0035

    0.0277

    0.04804

    Bias

    0.0049

    0.0833

    0.0017

    Variance

    0.0035

    0.0207

    0.04803

    SCI

    [5.1023,7.6421]

    [-0.1075,0.0826]

    [0.4927,0.6524]

    Length SCI

    2.5398

    0.1901

    0.1597

    Table 4: Progressively type I interval censored with random removal via the, MCMC procedure developed (Informative Priors).

    Different Parameters

    λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW gaaa@3838@

    β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@

    π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiWda haaa@3841@

    Average

    4.671

    0.0398

    0.6501

    MSE

    0.0673

    0.0559

    0.0656

    Bias

    0.0174

    0.0304

    0.0093

    Variance

    0.067

    0.0549

    0.0655

    SCI

    [4.8821,8.0307]

    [-0.1010,0.0721]

    [0.3881,0.7061]

    Length SCI

    3.1486

    0.1731

    0.318

    Table 5:Progressively type I interval censored with random removal via the, MCMC procedure developed (Non-Informative Priors).

    Both of density functions of π * ( λ/x,β,π ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWn aaCaaabeqcfasaaiaacQcaaaqcfa4aaeWaaeaacqaH7oaBcaGGVaGa amiEaiaacYcacqaHYoGycaGGSaGaeqiWdahacaGLOaGaayzkaaaaaa@436D@ and π * ( β/x,λ,π ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWn aaCaaabeqcfasaaiaacQcaaaqcfa4aaeWaaeaacqaHYoGycaGGVaGa amiEaiaacYcacqaH7oaBcaGGSaGaeqiWdahacaGLOaGaayzkaaaaaa@436D@ can be approximated by normal distribution functions but density function of π * ( π/x,λ,β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWn aaCaaabeqcfasaaiaacQcaaaqcfa4aaeWaaeaacqaHapaCcaGGVaGa amiEaiaacYcacqaH7oaBcaGGSaGaeqOSdigacaGLOaGaayzkaaaaaa@436D@ will be beta as mentioned in subsection (3.3) which are plotted in (Figure 1& 2) Chain of MCMC outputs of λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSb aa@382D@ , β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIb aa@381A@ and π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@ , using 100 000 MCMC samples. This was done with 1000 bootstrap sample and 100 000 MCMC sample and discard the first 50000 values as ‘burn-in’. The Bayes estimators can be seen to have the smaller risks than classical estimators for all the considered cases. It may also be noted that the Bayes estimators obtained under informative prior are more efficient than those obtained under non-informative priors. This indicates that the Bayesian procedure with accurate prior information provides more precise estimates. Also, The Length of the SCI (using informative prior) is smaller than the Length of the ACI and ABCI.

    Figure 1:Posterior density function of λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSb aa@382D@ , β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIb aa@381A@ and π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@ .
    Figure 2: Chain of MCMC outputs of λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSb aa@382D@ , β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIb aa@381A@ and π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@ .

    Conclusion

    The methodology developed in this paper will be very useful to the researchers, engineers, statisticians and in the field of medical where such type of life test is needed and especially where the Weibull distribution is used. we have considered the problem of estimation for flexible Weibull distribution in the presence of Progressive Type-I Interval censored sample with Binomial removals. The scope of this censoring scheme in clinical trials has been discussed. We have found that Bayesian procedure provides estimates of the unknown parameters of flexible Weibull model with smaller MSE. The length of SCI is smaller than that of the ACI and ABCI. Applying the MCMC process through the application of the MH algorithm to deal with the Bayesian estimation for another lifetime distributions under type I progressive interval censoring with random removal could be a fruitful future research.

    Appendix

    The asymptotic variance-covariance matrix of the maximum likelihood estimators for parameters , and are given by elements of the inverse of the Fisher information matrix with random removal will be

    I( λ ^ , β ^ , π ^ )=[ I 1 ( λ ^ , β ^ ) 0 0 I 2 ( π ^ ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMeaca GGOaGafq4UdWMbaKaacaGGSaGafqOSdiMbaKaacaGGSaGafqiWdaNb aKaacaGGPaGaeyypa0ZaamWaaeaafaqabeGacaaabaGaamysamaaBa aajuaibaGaaGymaaqcfayabaGaaiikaiqbeU7aSzaajaGaaiilaiqb ek7aIzaajaGaaiykaaqaaiaaicdaaeaacaaIWaaabaGaamysamaaBa aajuaibaGaaGOmaaqcfayabaGaaiikaiqbec8aWzaajaGaaiykaaaa aiaawUfacaGLDbaaaaa@512F@ ,

    Unfortunately, the exact mathematical expressions for the above expectations are very difficult to obtain. Therefore, we give the approximate (observed) asymptotic varaince-covariance matrix for the maximum likelihood estimators, which is obtained by dropping the expectation operator E, where

    I 1 1 ( λ ^ , β ^ )= [ ( 2 lnL(π) λ 2 ) ( 2 lnL(π) λβ ) ( 2 lnL(π) λβ ) ( 2 lnL(π) β 2 ) ] 1 λ= λ ^ 2 β= β ^ [ V( λ ^ ) Cov( λ ^ , β ^ ) Cov( λ ^ , β ^ ) V( β ^ ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMeada qhaaqcfasaaiaaigdaaeaacqGHsislcaaIXaaaaKqbakaacIcacuaH 7oaBgaqcaiaacYcacuaHYoGygaqcaiaacMcacqGH9aqpdaWadaqaau aabeqaciaaaeaadaqadaqaaiabgkHiTmaalaaabaGaeyOaIy7aaWba aeqajuaibaGaaGOmaaaajuaGciGGSbGaaiOBaiaadYeacaGGOaGaeq iWdaNaaiykaaqaaiabgkGi2kabeU7aSnaaCaaabeqaaiaaikdaaaaa aaGaayjkaiaawMcaaaqaamaabmaabaGaeyOeI0YaaSaaaeaacqGHci ITdaahaaqabKqbGeaacaaIYaaaaKqbakGacYgacaGGUbGaamitaiaa cIcacqaHapaCcaGGPaaabaGaeyOaIyRaeq4UdWMaeyOaIyRaeqOSdi gaaaGaayjkaiaawMcaaaqaamaabmaabaGaeyOeI0YaaSaaaeaacqGH ciITdaahaaqabKqbGeaacaaIYaaaaKqbakGacYgacaGGUbGaamitai aacIcacqaHapaCcaGGPaaabaGaeyOaIyRaeq4UdWMaeyOaIyRaeqOS digaaaGaayjkaiaawMcaaaqaamaabmaabaGaeyOeI0YaaSaaaeaacq GHciITdaahaaqabKqbGeaacaaIYaaaaKqbakGacYgacaGGUbGaamit aiaacIcacqaHapaCcaGGPaaabaGaeyOaIyRaeqOSdi2aaWbaaeqaju aibaGaaGOmaaaaaaaajuaGcaGLOaGaayzkaaaaaaGaay5waiaaw2fa amaaCaaabeqcfasaaiabgkHiTiaaigdaaaqcfa4aaSbaaeaacqaH7o aBcqGH9aqpcuaH7oaBgaqcamaaBaaajuaibaGaaGOmaaqcfayabaGa eqOSdiMaeyypa0JafqOSdiMbaKaaaeqaamaadmaabaqbaeqabiGaaa qaaiaadAfacaGGOaGafq4UdWMbaKaacaGGPaaabaGaam4qaiaad+ga caWG2bGaaiikaiqbeU7aSzaajaGaaiilaiqbek7aIzaajaGaaiykaa qaaiaadoeacaWGVbGaamODaiaacIcacuaH7oaBgaqcaiaacYcacuaH YoGygaqcaiaacMcaaeaacaWGwbGaaiikaiqbek7aIzaajaGaaiykaa aaaiaawUfacaGLDbaaaaa@AC8C@

    We explained how to find Fisher information matrix I 1 ( λ ^ , β ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMeada WgaaqcfasaaiaaigdaaKqbagqaaiaacIcacuaH7oaBgaqcaiaacYca cuaHYoGygaqcaiaacMcaaaa@3E5D@ in the Appendix B.

    I 2 ( π ^ )=E( 2 lnL(π) π 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMeada WgaaqcfasaaiaaikdaaKqbagqaaiaacIcacuaHapaCgaqcaiaacMca cqGH9aqpcaWGfbWaaeWaaeaacqGHsisldaWcaaqaaiabgkGi2oaaCa aabeqcfasaaiaaikdaaaqcfaOaciiBaiaac6gacaWGmbGaaiikaiab ec8aWjaacMcaaeaacqGHciITcqaHapaCdaahaaqabKqbGeaacaaIYa aaaaaaaKqbakaawIcacaGLPaaaaaa@4DE4@ ,

    and

    2 lnP(R) π 2 = 1 π 2 j=1 m1 r j 1 (1π) 2 [ ( m1 )( nm ) j=1 m1 ( mj ) r j ] . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeyOaIy7aaWbaaeqajuaibaGaaGOmaaaajuaGciGGSbGaaiOBaiaa dcfacaGGOaGaamOuaiaacMcaaeaacqGHciITcqaHapaCdaahaaqabK qbGeaacaaIYaaaaaaajuaGcqGH9aqpdaWcaaqaaiabgkHiTiaaigda aeaacqaHapaCdaahaaqabKqbGeaacaaIYaaaaaaajuaGdaaeWbqaai aadkhadaWgaaqcfasaaiaadQgaaKqbagqaaiabgkHiTmaalaaabaGa aGymaaqaaiaacIcacaaIXaGaeyOeI0IaeqiWdaNaaiykamaaCaaabe qcfasaaiaaikdaaaaaaKqbaoaadmaabaWaaeWaaeaacaWGTbGaeyOe I0IaaGymaaGaayjkaiaawMcaamaabmaabaGaamOBaiabgkHiTiaad2 gaaiaawIcacaGLPaaacqGHsisldaaeWbqaamaabmaabaGaamyBaiab gkHiTiaadQgaaiaawIcacaGLPaaacaWGYbWaaSbaaKqbGeaacaWGQb aajuaGbeaaaKqbGeaacaWGQbGaeyypa0JaaGymaaqaaiaad2gacqGH sislcaaIXaaajuaGcqGHris5aaGaay5waiaaw2faaaqcfasaaiaadQ gacqGH9aqpcaaIXaaabaGaamyBaiabgkHiTiaaigdaaKqbakabggHi LdGaaiOlaaaa@77A0@

    Numerical technique is needed to obtain the Fisher information matrix and the variance-covariance matrix. Note that under fixed and random removal the estimates based on intervals with equal length when the intervals are of equal length, so that monitoring and censoring occur periodically say T i =i.t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsfada WgaaqcfasaaiaadMgaaKqbagqaaiabg2da9iaadMgacaGGUaGaamiD aaaa@3CBC@ .

    We determine the second partials by differentiating the first partials, equations (8) and (9), obtaining

    2 logL λβ = i=1 m k i { 2 F( T i ) λβ 2 F( T i1 ) λβ F( T i )F( T i1 ) [ F( T i ) λ F( T i1 ) λ ][ F( T i ) β F( T i1 ) β ] [ F( T i )F( T i1 ) ] 2 }+ i=1 m r i { 2 λβ [ 1F( T i ) ] [ 1F( T i ) ] [ λ [ 1F( T i ) ] ][ β [ 1F( T i ) ] ] [ 1F( T i ) ] 2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWcaaWdaeaapeGaeyOaIy7damaaCaaabeqcfasaa8qacaaI YaaaaKqbakaadYgacaWGVbGaam4zaiaadYeaa8aabaWdbiabgkGi2k abeU7aSjabgkGi2kabek7aIbaacqGH9aqpdaaeWbqaaiaadUgadaWg aaqcfasaaiaadMgaaKqbagqaaaqcfasaaiaadMgacqGH9aqpcaaIXa aabaGaamyBaaqcfaOaeyyeIuoadaGadaWdaeaapeWaaSaaa8aabaWd bmaalaaapaqaa8qacqGHciITpaWaaWbaaeqajuaibaWdbiaaikdaaa qcfaOaamOramaabmaapaqaa8qacaWGubWdamaaBaaajuaibaWdbiaa dMgaaKqba+aabeaaa8qacaGLOaGaayzkaaaapaqaa8qacqGHciITcq aH7oaBcqGHciITcqaHYoGyaaGaeyOeI0YaaSaaa8aabaWdbiabgkGi 2+aadaahaaqabKqbGeaapeGaaGOmaaaajuaGcaWGgbWaaeWaa8aaba WdbiaadsfapaWaaSbaaKqbGeaapeGaamyAaiabgkHiTiaaigdaaKqb a+aabeaaa8qacaGLOaGaayzkaaaapaqaa8qacqGHciITcqaH7oaBcq GHciITcqaHYoGyaaaapaqaa8qacaWGgbWaaeWaa8aabaWdbiaadsfa paWaaSbaaKqbGeaapeGaamyAaaqcfa4daeqaaaWdbiaawIcacaGLPa aacqGHsislcaWGgbWaaeWaa8aabaWdbiaadsfapaWaaSbaaKqbGeaa peGaamyAaiabgkHiTiaaigdaaKqba+aabeaaa8qacaGLOaGaayzkaa aaaiabgkHiTmaalaaapaqaa8qadaWadaWdaeaapeWaaSaaa8aabaWd biabgkGi2kaadAeadaqadaWdaeaapeGaamiva8aadaWgaaqcfasaa8 qacaWGPbaajuaGpaqabaaapeGaayjkaiaawMcaaaWdaeaapeGaeyOa IyRaeq4UdWgaaiabgkHiTmaalaaapaqaa8qacqGHciITcaWGgbWaae Waa8aabaWdbiaadsfapaWaaSbaaKqbGeaapeGaamyAaiabgkHiTiaa igdaaKqba+aabeaaa8qacaGLOaGaayzkaaaapaqaa8qacqGHciITcq aH7oaBaaaacaGLBbGaayzxaaWaamWaa8aabaWdbmaalaaapaqaa8qa cqGHciITcaWGgbWaaeWaa8aabaWdbiaadsfapaWaaSbaaKqbGeaape GaamyAaaqcfa4daeqaaaWdbiaawIcacaGLPaaaa8aabaWdbiabgkGi 2kabek7aIbaacqGHsisldaWcaaWdaeaapeGaeyOaIyRaamOramaabm aapaqaa8qacaWGubWdamaaBaaajuaibaWdbiaadMgacqGHsislcaaI XaaajuaGpaqabaaapeGaayjkaiaawMcaaaWdaeaapeGaeyOaIyRaeq OSdigaaaGaay5waiaaw2faaaWdaeaapeWaamWaa8aabaWdbiaadAea daqadaWdaeaapeGaamiva8aadaWgaaqcfasaa8qacaWGPbaajuaGpa qabaaapeGaayjkaiaawMcaaiabgkHiTiaadAeadaqadaWdaeaapeGa amiva8aadaWgaaqcfasaa8qacaWGPbGaeyOeI0IaaGymaaqcfa4dae qaaaWdbiaawIcacaGLPaaaaiaawUfacaGLDbaapaWaaWbaaeqajuai baWdbiaaikdaaaaaaaqcfaOaay5Eaiaaw2haaiabgUcaRmaaqahaba GaamOCamaaBaaajuaibaGaamyAaaqcfayabaaajuaibaGaamyAaiab g2da9iaaigdaaeaacaWGTbaajuaGcqGHris5amaacmaapaqaa8qada WcaaWdaeaapeWaaSaaa8aabaWdbiabgkGi2+aadaahaaqabKqbGeaa peGaaGOmaaaaaKqba+aabaWdbiabgkGi2kabeU7aSjabgkGi2kabek 7aIbaadaWadaWdaeaapeGaaGymaiabgkHiTiaadAeadaqadaWdaeaa peGaamiva8aadaWgaaqcfasaa8qacaWGPbaajuaGpaqabaaapeGaay jkaiaawMcaaaGaay5waiaaw2faaaWdaeaapeWaamWaa8aabaWdbiaa igdacqGHsislcaWGgbWaaeWaa8aabaWdbiaadsfapaWaaSbaaKqbGe aapeGaamyAaaqcfa4daeqaaaWdbiaawIcacaGLPaaaaiaawUfacaGL DbaaaaGaeyOeI0YaaSaaa8aabaWdbmaadmaapaqaa8qadaWcaaWdae aapeGaeyOaIylapaqaa8qacqGHciITcqaH7oaBaaWaamWaa8aabaWd biaaigdacqGHsislcaWGgbWaaeWaa8aabaWdbiaadsfapaWaaSbaaK qbGeaapeGaamyAaaqcfa4daeqaaaWdbiaawIcacaGLPaaaaiaawUfa caGLDbaaaiaawUfacaGLDbaadaWadaWdaeaapeWaaSaaa8aabaWdbi abgkGi2cWdaeaapeGaeyOaIyRaeqOSdigaamaadmaapaqaa8qacaaI XaGaeyOeI0IaamOramaabmaapaqaa8qacaWGubWdamaaBaaajuaiba WdbiaadMgaaKqba+aabeaaa8qacaGLOaGaayzkaaaacaGLBbGaayzx aaaacaGLBbGaayzxaaaapaqaa8qadaWadaWdaeaapeGaaGymaiabgk HiTiaadAeadaqadaWdaeaapeGaamiva8aadaWgaaqcfasaa8qacaWG PbaajuaGpaqabaaapeGaayjkaiaawMcaaaGaay5waiaaw2faa8aada ahaaqabKqbGeaapeGaaGOmaaaaaaaajuaGcaGL7bGaayzFaaaaaa@1686@

    2 logL λ 2 = i=1 m k i { 2 F( T i ) λ 2 2 F( T i1 ) λ 2 F( T i )F( T i1 ) [ F( T i ) λ F( T i1 ) λ ] 2 [ F( T i )F( T i1 ) ] 2 }+ i=1 m r i { 2 λ 2 [ 1F( T i ) ] [ 1F( T i ) ] [ λ [ 1F( T i ) ] ] 2 [ 1F( T i ) ] 2 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWcaaWdaeaapeGaeyOaIy7damaaCaaabeqcfasaa8qacaaI YaaaaKqbakaadYgacaWGVbGaam4zaiaadYeaa8aabaWdbiabgkGi2k abeU7aS9aadaahaaqabKqbGeaapeGaaGOmaaaaaaqcfaOaeyypa0Za aabCaeaacaWGRbWaaSbaaKqbGeaacaWGPbaajuaGbeaaaKqbGeaaca WGPbGaeyypa0JaaGymaaqaaiaad2gaaKqbakabggHiLdWaaiWaa8aa baWdbmaalaaapaqaa8qadaWcaaWdaeaapeGaeyOaIy7damaaCaaabe qcfasaa8qacaaIYaaaaKqbakaadAeadaqadaWdaeaapeGaamiva8aa daWgaaqcfasaa8qacaWGPbaajuaGpaqabaaapeGaayjkaiaawMcaaa WdaeaapeGaeyOaIyRaeq4UdW2damaaCaaabeqcfasaa8qacaaIYaaa aaaajuaGcqGHsisldaWcaaWdaeaapeGaeyOaIy7damaaCaaabeqcfa saa8qacaaIYaaaaKqbakaadAeadaqadaWdaeaapeGaamiva8aadaWg aaqcfasaa8qacaWGPbGaeyOeI0IaaGymaaqcfa4daeqaaaWdbiaawI cacaGLPaaaa8aabaWdbiabgkGi2kabeU7aS9aadaahaaqabKqbGeaa peGaaGOmaaaaaaaajuaGpaqaa8qacaWGgbWaaeWaa8aabaWdbiaads fapaWaaSbaaKqbGeaapeGaamyAaaqcfa4daeqaaaWdbiaawIcacaGL PaaacqGHsislcaWGgbWaaeWaa8aabaWdbiaadsfapaWaaSbaaKqbGe aapeGaamyAaiabgkHiTiaaigdaaKqba+aabeaaa8qacaGLOaGaayzk aaaaaiabgkHiTmaalaaapaqaa8qadaWadaWdaeaapeWaaSaaa8aaba WdbiabgkGi2kaadAeadaqadaWdaeaapeGaamiva8aadaWgaaqcfasa a8qacaWGPbaajuaGpaqabaaapeGaayjkaiaawMcaaaWdaeaapeGaey OaIyRaeq4UdWgaaiabgkHiTmaalaaapaqaa8qacqGHciITcaWGgbWa aeWaa8aabaWdbiaadsfapaWaaSbaaKqbGeaapeGaamyAaiabgkHiTi aaigdaaKqba+aabeaaa8qacaGLOaGaayzkaaaapaqaa8qacqGHciIT cqaH7oaBaaaacaGLBbGaayzxaaWdamaaCaaabeqcfasaa8qacaaIYa aaaaqcfa4daeaapeWaamWaa8aabaWdbiaadAeadaqadaWdaeaapeGa amiva8aadaWgaaqcfasaa8qacaWGPbaajuaGpaqabaaapeGaayjkai aawMcaaiabgkHiTiaadAeadaqadaWdaeaapeGaamiva8aadaWgaaqc fasaa8qacaWGPbGaeyOeI0IaaGymaaqcfa4daeqaaaWdbiaawIcaca GLPaaaaiaawUfacaGLDbaapaWaaWbaaeqajuaibaWdbiaaikdaaaaa aaqcfaOaay5Eaiaaw2haaiabgUcaRmaawahabeqcfaYdaeaapeGaam yAaiabg2da9iaaigdaa8aabaWdbiaad2gaaKqba+aabaWdbiabggHi LdaacaWGYbWdamaaBaaajuaibaWdbiaadMgaaKqba+aabeaapeWaai Waa8aabaWdbmaalaaapaqaa8qadaWcaaWdaeaapeGaeyOaIy7damaa Caaabeqcfasaa8qacaaIYaaaaaqcfa4daeaapeGaeyOaIyRaeq4UdW 2damaaCaaabeqcfasaa8qacaaIYaaaaaaajuaGdaWadaWdaeaapeGa aGymaiabgkHiTiaadAeadaqadaWdaeaapeGaamiva8aadaWgaaqcfa saa8qacaWGPbaajuaGpaqabaaapeGaayjkaiaawMcaaaGaay5waiaa w2faaaWdaeaapeWaamWaa8aabaWdbiaaigdacqGHsislcaWGgbWaae Waa8aabaWdbiaadsfapaWaaSbaaKqbGeaapeGaamyAaaqcfa4daeqa aaWdbiaawIcacaGLPaaaaiaawUfacaGLDbaaaaGaeyOeI0YaaSaaa8 aabaWdbmaadmaapaqaa8qadaWcaaWdaeaapeGaeyOaIylapaqaa8qa cqGHciITcqaH7oaBaaWaamWaa8aabaWdbiaaigdacqGHsislcaWGgb WaaeWaa8aabaWdbiaadsfapaWaaSbaaKqbGeaapeGaamyAaaqcfa4d aeqaaaWdbiaawIcacaGLPaaaaiaawUfacaGLDbaaaiaawUfacaGLDb aapaWaaWbaaeqajuaibaWdbiaaikdaaaaajuaGpaqaa8qadaWadaWd aeaapeGaaGymaiabgkHiTiaadAeadaqadaWdaeaapeGaamiva8aada Wgaaqcfasaa8qacaWGPbaajuaGpaqabaaapeGaayjkaiaawMcaaaGa ay5waiaaw2faa8aadaahaaqabKqbGeaapeGaaGOmaaaaaaaajuaGca GL7bGaayzFaaaaaa@ECC4@

    2 logL β 2 = i=1 m k i { 2 F( T i ) β 2 2 F( T i1 ) β 2 F( T i )F( T i1 ) [ F( T i ) β F( T i1 ) β ] [ F( T i )F( T i1 ) ] 2 2 }+ i=1 m r i { 2 β 2 [ 1F( T i ) ] [ 1F( T i ) ] [ β [ 1F( T i ) ] ] [ 1F( T i ) ] 2 2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeyOaIy7aaWbaaeqajuaibaGaaGOmaaaajuaGciGGSbGaai4Baiaa cEgacaWGmbaabaGaeyOaIyRaeqOSdi2aaWbaaeqajuaibaGaaGOmaa aaaaqcfaOaeyypa0ZaaabCaeaacaWGRbWaaSbaaKqbGeaacaWGPbaa juaGbeaaaKqbGeaacaWGPbGaeyypa0JaaGymaaqaaiaad2gaaKqbak abggHiLdWaaiWaaeaadaWcaaqaamaalaaabaGaeyOaIy7aaWbaaeqa juaibaGaaGOmaaaajuaGcaWGgbWaaeWaaeaacaWGubWaaSbaaeaaca WGPbaabeaaaiaawIcacaGLPaaaaeaacqGHciITcqaHYoGydaahaaqa bKqbGeaacaaIYaaaaaaajuaGcqGHsisldaWcaaqaaiabgkGi2oaaCa aabeqaaiaaikdaaaGaamOramaabmaabaGaamivamaaBaaajuaibaGa amyAaiabgkHiTiaaigdaaKqbagqaaaGaayjkaiaawMcaaaqaaiabgk Gi2kabek7aInaaCaaabeqaaiaaikdaaaaaaaqaaiaadAeadaqadaqa aiaadsfadaWgaaqcfasaaiaadMgaaKqbagqaaaGaayjkaiaawMcaai abgkHiTiaadAeadaqadaqaaiaadsfadaWgaaqcfasaaiaadMgacqGH sislcaaIXaaajuaGbeaaaiaawIcacaGLPaaaaaGaeyOeI0YaaSaaae aadaWadaqaamaalaaabaGaeyOaIyRaamOramaabmaabaGaamivamaa BaaajuaibaGaamyAaaqcfayabaaacaGLOaGaayzkaaaabaGaeyOaIy RaeqOSdigaaiabgkHiTmaalaaabaGaeyOaIyRaamOramaabmaabaGa amivamaaBaaajuaibaGaamyAaiabgkHiTiaaigdaaKqbagqaaaGaay jkaiaawMcaaaqaaiabgkGi2kabek7aIbaaaiaawUfacaGLDbaaaeaa daWadaqaaiaadAeadaqadaqaaiaadsfadaWgaaqcfasaaiaadMgaaK qbagqaaaGaayjkaiaawMcaaiabgkHiTiaadAeadaqadaqaaiaadsfa daWgaaqcfasaaiaadMgacqGHsislcaaIXaaajuaGbeaaaiaawIcaca GLPaaaaiaawUfacaGLDbaadaahaaqabKqbGeaacaaIYaaaaaaajuaG daahaaqabKqbGeaacaaIYaaaaaqcfaOaay5Eaiaaw2haaiabgUcaRm aaqahabaGaamOCamaaBaaajuaibaGaamyAaaqcfayabaaajuaibaGa amyAaiabg2da9iaaigdaaeaacaWGTbaajuaGcqGHris5amaacmaaba WaaSaaaeaadaWcaaqaaiabgkGi2oaaCaaabeqcfasaaiaaikdaaaaa juaGbaGaeyOaIyRaeqOSdi2aaWbaaeqajuaibaGaaGOmaaaaaaqcfa 4aamWaaeaacaaIXaGaeyOeI0IaamOramaabmaabaGaamivamaaBaaa juaibaGaamyAaaqcfayabaaacaGLOaGaayzkaaaacaGLBbGaayzxaa aabaWaamWaaeaacaaIXaGaeyOeI0IaamOramaabmaabaGaamivamaa BaaajuaibaGaamyAaaqcfayabaaacaGLOaGaayzkaaaacaGLBbGaay zxaaaaaiabgkHiTmaalaaabaWaamWaaeaadaWcaaqaaiabgkGi2cqa aiabgkGi2kabek7aIbaadaWadaqaaiaaigdacqGHsislcaWGgbWaae WaaeaacaWGubWaaSbaaKqbGeaacaWGPbaajuaGbeaaaiaawIcacaGL PaaaaiaawUfacaGLDbaaaiaawUfacaGLDbaaaeaadaWadaqaaiaaig dacqGHsislcaWGgbWaaeWaaeaacaWGubWaaSbaaKqbGeaacaWGPbaa juaGbeaaaiaawIcacaGLPaaaaiaawUfacaGLDbaadaahaaqabKqbGe aacaaIYaaaaaaajuaGdaahaaqabKqbGeaacaaIYaaaaaqcfaOaay5E aiaaw2haaaaa@DFAA@

    F( T i )=1 e e λ T i β T i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAeada qadaqaaiaadsfadaWgaaqcfasaaiaadMgaaKqbagqaaaGaayjkaiaa wMcaaiabg2da9iaaigdacqGHsislcaWGLbWaaWbaaeqajuaibaGaey OeI0IaamyzaKqbaoaaCaaajuaibeqaaiabeU7aSjaadsfajuaGdaWg aaqcfasaaiaadMgaaeqaaaaacqGHsisljuaGdaWcaaqcfasaaiabek 7aIbqaaiaadsfajuaGdaWgaaqcfasaaiaadMgaaeqaaaaaaaaaaa@4C64@

    F( T i ) λ = 1 T i e λ T i β T i e e λ T i β T i = 1 T i e λ T i β T i { 1F( T i ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeyOaIyRaamOramaabmaabaGaamivamaaBaaajuaibaGaamyAaaqc fayabaaacaGLOaGaayzkaaaabaGaeyOaIyRaeq4UdWgaaiabg2da9m aalaaabaGaeyOeI0IaaGymaaqaaiaadsfadaWgaaqcfasaaiaadMga aKqbagqaaaaacaWGLbWaaWbaaeqajuaibaGaeq4UdWMaamivaKqbao aaBaaajuaibaGaamyAaiabgkHiTKqbaoaalaaajuaibaGaeqOSdiga baGaamivaKqbaoaaBaaajuaibaGaamyAaaqabaaaaaqabaaaaKqbak aadwgadaahaaqabKqbGeaacqGHsislcaWGLbqcfa4aaWbaaKqbGeqa baGaeq4UdWMaamivaKqbaoaaBaaajuaibaGaamyAaaqabaGaeyOeI0 scfa4aaSaaaKqbGeaacqaHYoGyaeaacaWGubqcfa4aaSbaaKqbGeaa caWGPbaabeaaaaaaaaaajuaGcqGH9aqpdaWcaaqaaiabgkHiTiaaig daaeaacaWGubWaaSbaaKqbGeaacaWGPbaajuaGbeaaaaGaamyzamaa CaaabeqcfasaaiabeU7aSjaadsfajuaGdaWgaaqcfasaaiaadMgacq GHsisljuaGdaWcaaqcfasaaiabek7aIbqaaiaadsfajuaGdaWgaaqc fasaaiaadMgaaeqaaaaaaeqaaaaajuaGdaGadaqaaiaaigdacqGHsi slcaWGgbWaaeWaaeaacaWGubWaaSbaaKqbGeaacaWGPbaajuaGbeaa aiaawIcacaGLPaaaaiaawUhacaGL9baaaaa@7AA5@

    F( T i ) β = 1 T i e λ T i β T i e e λ T i β T i = 1 T i e λ T i β T i { 1F( T i ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeyOaIyRaamOramaabmaabaGaamivamaaBaaajuaibaGaamyAaaqc fayabaaacaGLOaGaayzkaaaabaGaeyOaIyRaeqOSdigaaiabg2da9m aalaaabaGaeyOeI0IaaGymaaqaaiaadsfadaWgaaqcfasaaiaadMga aKqbagqaaaaacaWGLbWaaWbaaeqajuaibaGaeq4UdWMaamivaKqbao aaBaaajuaibaGaamyAaiabgkHiTKqbaoaalaaajuaibaGaeqOSdiga baGaamivaKqbaoaaBaaajuaibaGaamyAaaqabaaaaaqabaaaaKqbak aadwgadaahaaqabKqbGeaacqGHsislcaWGLbqcfa4aaWbaaKqbGeqa baGaeq4UdWMaamivaKqbaoaaBaaajuaibaGaamyAaaqabaGaeyOeI0 scfa4aaSaaaKqbGeaacqaHYoGyaeaacaWGubqcfa4aaSbaaKqbGeaa caWGPbaabeaaaaaaaaaajuaGcqGH9aqpdaWcaaqaaiabgkHiTiaaig daaeaacaWGubWaaSbaaKqbGeaacaWGPbaajuaGbeaaaaGaamyzamaa CaaabeqcfasaaiabeU7aSjaadsfajuaGdaWgaaqcfasaaiaadMgacq GHsisljuaGdaWcaaqcfasaaiabek7aIbqaaiaadsfajuaGdaWgaaqc fasaaiaadMgaaeqaaaaaaeqaaaaajuaGdaGadaqaaiaaigdacqGHsi slcaWGgbWaaeWaaeaacaWGubWaaSbaaKqbGeaacaWGPbaajuaGbeaa aiaawIcacaGLPaaaaiaawUhacaGL9baaaaa@7A92@

    F( T i ) λ =[ 1F( T i ) ]= F( T i ) λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeyOaIyRaamOramaabmaabaGaamivamaaBaaajuaibaGaamyAaaqc fayabaaacaGLOaGaayzkaaaabaGaeyOaIyRaeq4UdWgaaiabg2da9m aadmaabaGaaGymaiabgkHiTiaadAeadaqadaqaaiaadsfadaWgaaqc fasaaiaadMgaaKqbagqaaaGaayjkaiaawMcaaaGaay5waiaaw2faai abg2da9iabgkHiTmaalaaabaGaeyOaIyRaamOramaabmaabaGaamiv amaaBaaajuaibaGaamyAaaqcfayabaaacaGLOaGaayzkaaaabaGaey OaIyRaeq4UdWgaaaaa@5514@

    F( T i ) β =[ 1F( T i ) ]= F( T i ) β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeyOaIyRaamOramaabmaabaGaamivamaaBaaajuaibaGaamyAaaqc fayabaaacaGLOaGaayzkaaaabaGaeyOaIyRaeqOSdigaaiabg2da9m aadmaabaGaaGymaiabgkHiTiaadAeadaqadaqaaiaadsfadaWgaaqc fasaaiaadMgaaKqbagqaaaGaayjkaiaawMcaaaGaay5waiaaw2faai abg2da9iabgkHiTmaalaaabaGaeyOaIyRaamOramaabmaabaGaamiv amaaBaaajuaibaGaamyAaaqcfayabaaacaGLOaGaayzkaaaabaGaey OaIyRaeqOSdigaaaaa@54EE@

    2 F( T i ) λ 2 = 1 T i e λ T i β T i { T i [ 1F( T i ) ]+ F( T i ) λ } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeyOaIy7aaWbaaeqajuaibaGaaGOmaaaajuaGcaWGgbWaaeWaaeaa caWGubWaaSbaaKqbGeaacaWGPbaajuaGbeaaaiaawIcacaGLPaaaae aacqGHciITcqaH7oaBdaahaaqabKqbGeaacaaIYaaaaaaajuaGcqGH 9aqpdaWcaaqaaiaaigdaaeaacaWGubWaaSbaaKqbGeaacaWGPbaaju aGbeaaaaGaamyzamaaCaaabeqcfasaaiabeU7aSjaadsfajuaGdaWg aaqcfasaaiaadMgaaeqaaiabgkHiTKqbaoaalaaajuaibaGaeqOSdi gabaGaamivaKqbaoaaBaaajuaibaGaamyAaaqabaaaaaaajuaGdaGa daqaaiaadsfadaWgaaqcfasaaiaadMgaaKqbagqaamaadmaabaGaaG ymaiabgkHiTiaadAeadaqadaqaaiaadsfadaWgaaqcfasaaiaadMga aKqbagqaaaGaayjkaiaawMcaaaGaay5waiaaw2faaiabgUcaRmaala aabaGaeyOaIyRaamOramaabmaabaGaamivamaaBaaajuaibaGaamyA aaqcfayabaaacaGLOaGaayzkaaaabaGaeyOaIyRaeq4UdWgaaaGaay 5Eaiaaw2haaaaa@6B99@

    2 F( T i ) β 2 = 1 T i e λ T i β T i { 1 T i [ 1F( T i ) ]+ F( T i ) β } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeyOaIy7aaWbaaeqajuaibaGaaGOmaaaajuaGcaWGgbWaaeWaaeaa caWGubWaaSbaaKqbGeaacaWGPbaajuaGbeaaaiaawIcacaGLPaaaae aacqGHciITcqaHYoGydaahaaqabKqbGeaacaaIYaaaaaaajuaGcqGH 9aqpdaWcaaqaaiaaigdaaeaacaWGubWaaSbaaKqbGeaacaWGPbaaju aGbeaaaaGaamyzamaaCaaabeqcfasaaiabeU7aSjaadsfajuaGdaWg aaqcfasaaiaadMgaaeqaaiabgkHiTKqbaoaalaaajuaibaGaeqOSdi gabaGaamivaKqbaoaaBaaajuaibaGaamyAaaqabaaaaaaajuaGdaGa daqaamaalaaabaGaaGymaaqaaiaadsfadaWgaaqcfasaaiaadMgaaK qbagqaaaaadaWadaqaaiaaigdacqGHsislcaWGgbWaaeWaaeaacaWG ubWaaSbaaKqbGeaacaWGPbaajuaGbeaaaiaawIcacaGLPaaaaiaawU facaGLDbaacqGHRaWkdaWcaaqaaiabgkGi2kaadAeadaqadaqaaiaa dsfadaWgaaqcfasaaiaadMgaaKqbagqaaaGaayjkaiaawMcaaaqaai abgkGi2kabek7aIbaaaiaawUhacaGL9baaaaa@6C3E@

    2 F( T i ) λβ = e λ T i β T i { [ 1F( T i ) ]+ T i F( T i ) β } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeyOaIy7aaWbaaeqajuaibaGaaGOmaaaajuaGcaWGgbWaaeWaaeaa caWGubWaaSbaaKqbGeaacaWGPbaajuaGbeaaaiaawIcacaGLPaaaae aacqGHciITcqaH7oaBcqGHciITcqaHYoGyaaGaeyypa0JaeyOeI0Ia amyzamaaCaaabeqcfasaaiabeU7aSjaadsfajuaGdaWgaaqcfasaai aadMgaaeqaaiabgkHiTKqbaoaalaaajuaibaGaeqOSdigabaGaamiv aKqbaoaaBaaajuaibaGaamyAaaqabaaaaaaajuaGdaGadaqaamaadm aabaGaaGymaiabgkHiTiaadAeadaqadaqaaiaadsfadaWgaaqcfasa aiaadMgaaKqbagqaaaGaayjkaiaawMcaaaGaay5waiaaw2faaiabgU caRiaadsfadaWgaaqcfasaaiaadMgaaKqbagqaamaalaaabaGaeyOa IyRaamOramaabmaabaGaamivamaaBaaajuaibaGaamyAaaqcfayaba aacaGLOaGaayzkaaaabaGaeyOaIyRaeqOSdigaaaGaay5Eaiaaw2ha aaaa@6A71@

    Note that: T i F( T i ) β = 1 T i F( T i ) λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsfada WgaaqcfasaaiaadMgaaKqbagqaamaalaaabaGaeyOaIyRaamOramaa bmaabaGaamivamaaBaaajuaibaGaamyAaaqcfayabaaacaGLOaGaay zkaaaabaGaeyOaIyRaeqOSdigaaiabg2da9iabgkHiTmaalaaabaGa aGymaaqaaiaadsfadaWgaaqcfasaaiaadMgaaKqbagqaaaaadaWcaa qaaiabgkGi2kaadAeadaqadaqaaiaadsfadaWgaaqcfasaaiaadMga aKqbagqaaaGaayjkaiaawMcaaaqaaiabgkGi2kabeU7aSbaaaaa@517C@

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