ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 4 Issue 2 - 2016
On Discrete Three Parameter Burr Type XII and Discrete Lomax Distributions and Their Applications to Model Count Data from Medical Science
Para BA* and Jan TR
Department of statistics, University of Kashmir, India
Received: May 19, 2016 | Published: July 23, 2016
*Corresponding author: Bilal Ahmad Para, Department of statistics, University of Kashmir, Srinagar, J&K(India)-192301, Srinagar, Jammu and Kashmir, India, Tel: 09797056761; Email:
Citation: Para BA, Jan TR (2016) On Discrete Three Parameter Burr Type XII and Discrete Lomax Distributions and Their Applications to Model Count Data from Medical Science. Biom Biostat Int J 4(2): 00092. DOI: 10.15406/bbij.2016.04.00092

Abstract

In this paper we propose a discrete analogue of three parameter Burr type XII distribution and discrete Lomax distribution as new discrete models using the general approach of discretization of continuous distribution. The models are plausible in modeling discrete data and exhibit both increasing and decreasing hazard rates. We shall first study some basic distributional and moment properties of these new distributions. Then, certain structural properties of the distributions such as their unimodality, hazard rate behaviors and the second rate of failure functions are discussed. Developing a discrete versions of three parameter Burr type XII and Lomax distributions would be helpful in modeling a discrete data which exhibits heavy tails and can be useful in medical science and other fields. The equivalence of discrete three parameter Burr type XII (DBD-XII) and continuous Burr type XII (BD-XII) distributions has been established and similarly characterization results have also been made to establish a direct link between the discrete Lomax distribution and its continuous counterpart. Various theorems relating a three parameter discrete Burr type XII distribution and discrete Lomax distribution with other statistical distributions have also been proved. Finally, the models are examined with an example data set originated from a study [1,2], data set of counts of cysts of kidneys using steroids and compared with the classical models.

Keywords: Discrete Lomax Distribution; AIC; ML estimate; Failure rate; Medical Sciences; Index of Dispersion

Introduction

Statistical models describe a phenomenon in the form of mathematical equations. Plethora of continuous lifetime models in reliability theory is now available in the subject to portray the survival behavior of a component or a system. Most of the lifetimes are continuous in nature and hence many continuous life distributions have been studied in literature Kapur & Lamberson [3], Lawless [4] and Sinha [5]. However, it is sometimes impossible or inconvenient in life testing experiments to measure the life length of a device on a continuous scale. Equipment or a piece of equipment operates in cycles and experimenter observes the number of cycles successfully completed prior to failure. A frequently referred example is copier whose life length would be the total number of copies it produces. Another example is the lifetime of an on/off switching device is a discrete random variable, or life length of a device receiving a number of shocks it sustain before it fails. Or in case of survival analysis, we may record the number of days of survival for lung cancer patients since therapy, or the times from remission to relapse are also usually recorded in number of days. In the recent past special roles of discrete distribution is getting recognition from the analysts in the field of reliability theory. In this context, the well known distributions namely geometric and negative binomial are known discrete alternatives for the exponential and gamma distributions, respectively. It is also well known that these discrete distributions have monotonic hazard rate functions and thus they are unsuitable for some situations. Fortunately, many continuous distributions can be discretized. As mentioned earlier, the discrete versions of exponential and gamma are geometric and negative binomial. There are three discrete versions of the continuous Weibull distribution [14]. The discrete versions of the normal and rayleigh distributions were also proposed by Roy [6,7]. Discrete analogues of two parameter Burr XII and Pareto distributions were also proposed by Krishna & Punder [8]. Recently discrete inverse Weibull distribution was studied [9], which is a discrete version of the continuous inverse Weibull variable, defined as X 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGybWaaWbaaeqajuaibaGaeyOeI0IaaGymaaaaaaa@3979@ where X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwaa aa@3761@  denotes the continuous Weibull random variable. Para & Jan [10] proposed a discrete version of two parameter Burr type III distribution as a reliability model to fit a range of discrete life time data. Deniz & Ojeda [11] introduced a discrete version of Lindley distribution by discretizing the continuous failure model of the Lindley distribution. Also, a compound discrete Lindley distribution in closed form is obtained after revising some of its properties. Nekoukhou et al. [12] presented a discrete analog of the generalized exponential distribution, which can be viewed as another generalization of the geometric distribution, and some of its distributional and moment properties were discussed.

 In the present paper we propose a three parameter discrete Burr type XII (DBD-XII) model and a two parameter discrete Lomax model as there is a need to find more plausible discrete life time distributions or survival models in medical science and other fields, to fit to various life time data. The model has a flexible index of dispersion which broaden its range to fit a data sets arising in medical science/biological science, engineering, finance etc.

Burr [13] introduced a family of distributions includes twelve types of cumulative distribution functions, which yield a variety of density shapes. The two important members of the family are Burr type III and Burr type XII distributions. Types III and XII are the simplest functionally and therefore, the two distributions are the most desirable for statistical modeling.

 A continuous random variable X is said to follow a three parameter Burr type XII distribution if its pdf is given by

f(x)= { ck γ ( x y ) c1 0 ( 1+ ( x y ) c ) ( k+1 ) , x>0,c>0,k>o,γ>0 elsewhere MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzai aacIcacaWG4bGaaiykaiabg2da9maaceaabaWaaSaaaeaacaWGJbGa am4Aaaqaaiabeo7aNbaadaqadaqaamaalaaabaGaamiEaaqaaiaadM haaaaacaGLOaGaayzkaaaacaGL7baadaahaaqabKqbGeaacaWGJbGa eyOeI0IaaGymaaaajuaGdaWgaaqcfasaaiaaicdaaKqbagqaamaabm aabaGaaGymaiabgUcaRmaabmaabaWaaSaaaeaacaWG4baabaGaamyE aaaaaiaawIcacaGLPaaadaahaaqabKqbGeaacaWGJbaaaaqcfaOaay jkaiaawMcaamaaCaaabeqcfasaaiabgkHiTKqbaoaabmaajuaibaGa am4AaiabgUcaRiaaigdaaiaawIcacaGLPaaaaaqcfaOaaiilamaawa fabeqcfasaaiaadwgacaWGSbGaam4CaiaadwgacaWG3bGaamiAaiaa dwgacaWGYbGaamyzaaqcfayabeaacaWG4bGaeyOpa4JaaGimaiaacY cacaWGJbGaeyOpa4JaaGimaiaacYcacaWGRbGaeyOpa4Jaam4Baiaa cYcacqaHZoWzcqGH+aGpcaaIWaaaaaaa@6F44@

and its cumulative distribution function is given by

F( x ) =1 ( 1+ ( x γ ) c ) k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGgbWaaeWaa8aabaWdbiaabIhaaiaawIcacaGLPaaacaqG GcGaeyypa0JaaGymaiabgkHiTmaabmaapaqaa8qacaaIXaGaey4kaS YaaeWaa8aabaWdbmaalaaapaqaa8qacaqG4baapaqaa8qacaqGZoaa aaGaayjkaiaawMcaa8aadaahaaqabKqbGeaapeGaae4yaaaaaKqbak aawIcacaGLPaaapaWaaWbaaeqajuaibaWdbiabgkHiTiaabUgaaaaa aa@497E@

x>0,k>0,c>0,γ>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqG4bGaeyOpa4JaaGimaiaacYcacaWGRbGaeyOpa4JaaGim aiaacYcacaWGJbGaeyOpa4JaaGimaiaacYcacaqGZoGaeyOpa4JaaG imaaaa@43C9@

When c=1, the three parameter Burr type XII distribution becomes Lomax distribution with pdf given

f( x )={ k γ   ( 1+( x γ ) ) ( k+1 )   ,x>0 , k>0,γ>0 0                                                     elsewhere MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGMbWaaeWaa8aabaWdbiaabIhaaiaawIcacaGLPaaacqGH 9aqpdaGabaWdaeaafaqabeGabaaabaWdbmaalaaapaqaa8qacaqGRb aapaqaa8qacaqGZoaaaiaacckadaqadaWdaeaapeGaaGymaiabgUca Rmaabmaapaqaa8qadaWcaaWdaeaapeGaaeiEaaWdaeaapeGaae4Sda aaaiaawIcacaGLPaaaaiaawIcacaGLPaaapaWaaWbaaeqajuaibaWd biabgkHiTKqbaoaabmaajuaipaqaa8qacaqGRbGaey4kaSIaaGymaa GaayjkaiaawMcaaaaajuaGcaGGGcGaaiiOaiaacYcacaWG4bGaeyOp a4JaaGimaiaacckacaGGSaGaaiiOaiaadUgacqGH+aGpcaaIWaGaai ilaiabeo7aNjabg6da+iaaicdaa8aabaWdbiaaicdacaGGGcGaaiiO aiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGc GaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaaccka caGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOai aacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGa aiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckaca GGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaa cckacaGGGcGaaiiOaiaadwgacaWGSbGaam4CaiaadwgacaWG3bGaam iAaiaadwgacaWGYbGaamyzaaaaaiaawUhaaaaa@A396@

and its cumulative distribution function is given by

F( x ) =1 ( 1+( x γ ) ) k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGgbWaaeWaa8aabaWdbiaabIhaaiaawIcacaGLPaaacaqG GcGaeyypa0JaaGymaiabgkHiTmaabmaapaqaa8qacaaIXaGaey4kaS YaaeWaa8aabaWdbmaalaaapaqaa8qacaqG4baapaqaa8qacaqGZoaa aaGaayjkaiaawMcaaaGaayjkaiaawMcaa8aadaahaaqabKqbGeaape GaeyOeI0Iaae4Aaaaaaaa@479B@

x>0,k>0,γ>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqG4bGaeyOpa4JaaGimaiaacYcacaWGRbGaeyOpa4JaaGim aiaacYcacaqGZoGaeyOpa4JaaGimaaaa@406F@

Figures 1-4 gives the pdf plot for three parameter Burr type XII distribution and Lomax distribution for different values of parameters. Figures 3&4 are especially for Lomax distribution. It is evident that the distribution of the rv X exhibit a right skewed nature.

Figure 1: pdf plot for BD-XII (c,k,γ)

Figure 2: pdf plot for BD-XII (c,k,γ).

Figure 3: pdf plot for BD-XII (c,k, γ).

Figure 4: PDF plot for BD-XII (c, k, γ).

The various reliability measures of three parameter Burr type XII random variable X are given by

  1. Survival function

                s( x )=1 0 x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGZbWaaeWaa8aabaWdbiaabIhaaiaawIcacaGLPaaacqGH 9aqpcaaIXaGaeyOeI0YaaybCaeqajuaipaqaa8qacaaIWaaapaqaa8 qacaqG4baajuaGpaqaa8qacqGHRiI8aaGaaeOzamaabmaapaqaa8qa caqG4baacaGLOaGaayzkaaGaaeizaiaabIhaaaa@4781@

=1 0 x ck γ   ( x γ ) c1 ( 1+ ( x γ ) c ) ( k+1 ) dx MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGH9aqpcaaIXaGaeyOeI0YaaybCaeqajuaipaqaa8qacaaI Waaapaqaa8qacaqG4baajuaGpaqaa8qacqGHRiI8aaWaaSaaa8aaba WdbiaabogacaqGRbaapaqaa8qacaqGZoaaaiaabckadaqadaWdaeaa peWaaSaaa8aabaWdbiaabIhaa8aabaWdbiaabo7aaaaacaGLOaGaay zkaaWdamaaCaaabeqcfasaa8qacaqGJbGaeyOeI0IaaGymaaaajuaG daqadaWdaeaapeGaaGymaiabgUcaRmaabmaapaqaa8qadaWcaaWdae aapeGaaeiEaaWdaeaapeGaae4SdaaaaiaawIcacaGLPaaapaWaaWba aeqajuaibaWdbiaabogaaaaajuaGcaGLOaGaayzkaaWdamaaCaaabe qcfasaa8qacqGHsisljuaGdaqadaqcfaYdaeaapeGaae4AaiabgUca RiaaigdaaiaawIcacaGLPaaaaaqcfaOaaeizaiaabIhaaaa@5CBA@

=( 1+ ( x γ ) c ) γ k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGH9aqpdaqadaWdaeaapeGaaGymaiabgUcaRmaabmaapaqa a8qadaWcaaWdaeaapeGaaeiEaaWdaeaapeGaae4SdaaaaiaawIcaca GLPaaapaWaaWbaaeqajuaibaWdbiaabogaaaaajuaGcaGLOaGaayzk aaGaae4Sd8aadaahaaqabKqbGeaapeGaeyOeI0Iaae4Aaaaaaaa@4480@

                               > 0; c > 0; k > 0; γ>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqG4bGaaeiOaiabg6da+iaacckacaaIWaGaai4oaiaaccka caWGJbGaaiiOaiabg6da+iaacckacaaIWaGaai4oaiaacckacaWGRb GaaiiOaiabg6da+iaacckacaaIWaGaai4oaiaacckacaqGZoGaeyOp a4JaaGimaaaa@4E39@        

  1. The failure rate is given by

          r( x )=  ck γ   ( x γ ) c1 /1+ ( x γ ) c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGYbWaaeWaa8aabaWdbiaabIhaaiaawIcacaGLPaaacqGH 9aqpcaqGGcWaaSaaa8aabaWdbiaabogacaqGRbaapaqaa8qacaqGZo aaaiaabckadaqadaWdaeaapeWaaSaaa8aabaWdbiaabIhaa8aabaWd biaabo7aaaaacaGLOaGaayzkaaWdamaaCaaabeqcfasaa8qacaqGJb GaeyOeI0IaaGymaaaajuaGcaGGVaGaaGymaiabgUcaRmaabmaapaqa a8qadaWcaaWdaeaapeGaaeiEaaWdaeaapeGaae4SdaaaaiaawIcaca GLPaaapaWaaWbaaeqajuaibaWdbiaabogaaaaaaa@5068@

> 0; c > 0; k > 0; γ>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqG4bGaaeiOaiabg6da+iaacckacaaIWaGaai4oaiaaccka caWGJbGaaiiOaiabg6da+iaacckacaaIWaGaai4oaiaacckacaWGRb GaaiiOaiabg6da+iaacckacaaIWaGaai4oaiaacckacaqGZoGaeyOp a4JaaGimaaaa@4E39@

  1. The second rate of failure is given by

         SRF( x )=log( s( x ) s( x+1 ) )=klog( 1+ ( x/γ ) c 1+ ( ( x+1 )/γ ) c ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGtbGaaeOuaiaabAeadaqadaWdaeaapeGaaeiEaaGaayjk aiaawMcaaiabg2da9iGacYgacaGGVbGaai4zamaabmaapaqaa8qada WcaaWdaeaapeGaae4Camaabmaapaqaa8qacaqG4baacaGLOaGaayzk aaaapaqaa8qacaqGZbWaaeWaa8aabaWdbiaabIhacqGHRaWkcaaIXa aacaGLOaGaayzkaaaaaaGaayjkaiaawMcaaiabg2da9iabgkHiTiaa bUgacaqGSbGaae4BaiaabEgadaqadaWdaeaapeWaaSaaa8aabaWdbi aaigdacqGHRaWkdaqadaWdaeaapeGaaeiEaiaac+cacaqGZoaacaGL OaGaayzkaaWdamaaCaaabeqcfasaa8qacaqGJbaaaaqcfa4daeaape GaaGymaiabgUcaRmaabmaapaqaa8qadaqadaWdaeaapeGaaeiEaiab gUcaRiaaigdaaiaawIcacaGLPaaacaGGVaGaae4SdaGaayjkaiaawM caa8aadaahaaqabKqbGeaapeGaae4yaaaaaaaajuaGcaGLOaGaayzk aaaaaa@655C@

> 0; c > 0; k > 0; γ>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqG4bGaaeiOaiabg6da+iaacckacaaIWaGaai4oaiaaccka caWGJbGaaiiOaiabg6da+iaacckacaaIWaGaai4oaiaacckacaWGRb GaaiiOaiabg6da+iaacckacaaIWaGaai4oaiaacckacaqGZoGaeyOp a4JaaGimaaaa@4E39@

  1. The rth moment is
  2.     E( x r )= 0 x r f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGGcGaaeiOaiaabweadaqadaWdaeaapeGaaeiEa8aadaah aaqabKqbGeaapeGaaeOCaaaaaKqbakaawIcacaGLPaaacqGH9aqpda GfWbqabKqbG8aabaWdbiaaicdaa8aabaWdbiabe6HiLcqcfa4daeaa peGaey4kIipaaiaabIhapaWaaWbaaeqajuaibaWdbiaabkhaaaqcfa OaaeOzamaabmaapaqaa8qacaqG4baacaGLOaGaayzkaaGaaeizaiaa bIhaaaa@4D43@

    = r β( r c +1, k r c ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGH9aqpcaqGRbGaae4Sd8aadaahaaqabKqbGeaapeGaaeOC aaaajuaGcaqGYoWaaeWaa8aabaWdbmaalaaapaqaa8qacaqGYbaapa qaa8qacaqGJbaaaiabgUcaRiaaigdacaGGSaGaaeiOaiaabUgacqGH sisldaWcaaWdaeaapeGaaeOCaaWdaeaapeGaae4yaaaaaiaawIcaca GLPaaaaaa@4841@
 Where

β( a,b )= 0 x a1 (1+x) a+b dx MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGYoWaaeWaa8aabaWdbiaabggacaGGSaGaaeOyaaGaayjk aiaawMcaaiabg2da9maawahabeqcfaYdaeaapeGaaGimaaWdaeaape GaeqOhIukajuaGpaqaa8qacqGHRiI8aaWaaSaaa8aabaWdbiaabIha paWaaWbaaeqajuaibaWdbiaabggacqGHsislcaaIXaaaaaqcfa4dae aapeGaaiikaiaaigdacqGHRaWkcaqG4bGaaiyka8aadaahaaqabKqb GeaapeGaaeyyaiabgUcaRiaabkgaaaaaaKqbakaabsgacaqG4baaaa@50EC@  ,

> 0; c > 0; k > 0; σ>0;ck>r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqG4bGaaeiOaiabg6da+iaacckacaaIWaGaai4oaiaaccka caWGJbGaaiiOaiabg6da+iaacckacaaIWaGaai4oaiaacckacaWGRb GaaiiOaiabg6da+iaacckacaaIWaGaai4oaiaacckacqaHdpWCcqGH +aGpcaaIWaGaai4oaiaadogacaWGRbGaeyOpa4JaamOCaaaa@5359@

The convergence of the rth moment is only possible if ck>r

Three Parameter Discrete Burr Type XII and Discrete Lomax Model

Roy [14] pointed out that the univariate geometric distribution can be viewed as a discrete concentration of a corresponding exponential distribution in the following manner:

[ X=x ]=s( x ) s ( x+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGWbGaaeiOamaadmaapaqaa8qacaqGybGaeyypa0JaaeiE aaGaay5waiaaw2faaiabg2da9iaabohadaqadaWdaeaapeGaaeiEaa GaayjkaiaawMcaaiabgkHiTiaabckacaqGZbGaaeiOamaabmaapaqa a8qacaqG4bGaey4kaSIaaGymaaGaayjkaiaawMcaaaaa@4AB0@

 When x = 0, 1, 2,…..

Where X is discrete random variable following geometric distribution with probability mass functions as

( x )=  θ x ( 1θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGWbGaaeiOamaabmaapaqaa8qacaqG4baacaGLOaGaayzk aaGaeyypa0JaaeiOaiaabI7apaWaaWbaaeqajuaibaWdbiaabIhaaa qcfa4aaeWaa8aabaWdbiaaigdacqGHsislcaqG4oaacaGLOaGaayzk aaaaaa@454B@   x = 0,1,2,…….

Where s(x) represents the survival function of an exponential distribution of the form s( x ) = exp( λx ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGZbWdamaabmaabaWdbiaadIhaa8aacaGLOaGaayzkaaWd biaabccacqGH9aqpcaqGGaGaamyzaiaadIhacaWGWbWdamaabmaaba WdbiabgkHiTiabeU7aSjaadIhaa8aacaGLOaGaayzkaaaaaa@44DD@ clearly θ= exp( λ ), 0 < θ< 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqG4oGaeyypa0JaaeiiaiaadwgacaWG4bGaamiCa8aadaqa daqaa8qacqGHsislcqaH7oaBa8aacaGLOaGaayzkaaWdbiaacYcaca qGGaGaaGimaiaabccacqGH8aapcaGGGcGaeqiUdeNaeyipaWJaaeii aiaaigdaaaa@49C0@ .

Thus, one to one correspondence between the geometric distribution and the exponential distribution can be established, the survival functions being of the same form.

The general approach of dicretising a continuous variable is to introduce a greatest integer function of X i.e., [X] (the greatest integer less than or equal to X till it reaches the integer), in order to introduce grouping on a time axis.

A discrete Burr type XII variable, dX can be viewed as the discrete concentration of the continuous Burr type XII variable X, where the corresponding probability mass function of dX can be written as:

P( dX=x )=p( x )=s( x )s( x+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGqbWaaeWaa8aabaWdbiaabsgacaqGybGaeyypa0JaaeiE aaGaayjkaiaawMcaaiabg2da9iaabchadaqadaWdaeaapeGaaeiEaa GaayjkaiaawMcaaiabg2da9iaabohadaqadaWdaeaapeGaaeiEaaGa ayjkaiaawMcaaiabgkHiTiaabohadaqadaWdaeaapeGaaeiEaiabgU caRiaaigdaaiaawIcacaGLPaaaaaa@4C41@

The probability mass function takes the form

P( x )= β log( 1+ ( x/γ ) c ) β log( 1+ ( ( x+1 )/γ ) c )    x=0,1,2,3. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGqbWaaeWaa8aabaWdbiaabIhaaiaawIcacaGLPaaacqGH 9aqpcaqGYoWdamaaCaaabeqcfasaa8qaciGGSbGaai4BaiaacEgaju aGdaqadaqcfaYdaeaapeGaaGymaiabgUcaRKqbaoaabmaajuaipaqa a8qacaqG4bGaai4laiaabo7aaiaawIcacaGLPaaajuaGpaWaaWbaaK qbGeqabaWdbiaabogaaaaacaGLOaGaayzkaaaaaKqbakabgkHiTiaa bk7apaWaaWbaaeqajuaibaWdbiGacYgacaGGVbGaai4zaKqbaoaabm aajuaipaqaa8qacaaIXaGaey4kaSscfa4aaeWaaKqbG8aabaqcfa4d bmaabmaajuaipaqaa8qacaqG4bGaey4kaSIaaGymaaGaayjkaiaawM caaiaac+cacaqGZoaacaGLOaGaayzkaaqcfa4damaaCaaajuaibeqa a8qacaqGJbaaaaGaayjkaiaawMcaaaaajuaGcaqGGcGaaeiOaiaabc kacaqG4bGaeyypa0JaaGimaiaacYcacaaIXaGaaiilaiaaikdacaGG SaGaaG4maiabgAci8kaac6caaaa@6D1F@                                                 (3.1) 

Where β= e k ; 0<β<1;γ>0;c>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGYoGaeyypa0Jaaeyza8aadaahaaqabKqbGeaapeGaeyOe I0Iaae4AaaaajuaGcaGG7aGaaeiOaiaaicdacqGH8aapcqaHYoGycq GH8aapcaaIXaGaai4oaiabeo7aNjabg6da+iaaicdacaGG7aGaam4y aiabg6da+iaaicdaaaa@4B34@

And the cumulative distribution function is given by

F( x )=1 β log(1+ ((x+1)/γ) c ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGgbWdamaabmaabaWdbiaadIhaa8aacaGLOaGaayzkaaWd biabg2da9iaaigdacqGHsislcqaHYoGydaahaaqabKqbGeaaciGGSb Gaai4BaiaacEgacaGGOaGaaGymaiabgUcaRiaacIcacaGGOaGaamiE aiabgUcaRiaaigdacaGGPaGaai4laiabeo7aNjaacMcajuaGdaahaa qcfasabeaacaWGJbaaaiaacMcaaaaaaa@4E04@    Where β= e (k) ;0<β<1;γ>0;c>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHYoGycqGH9aqpcaWGLbWdamaaCaaabeqcfasaa8qacaGG OaGaeyOeI0Iaam4AaiaacMcaaaqcfaOaai4oaiaaicdacqGH8aapcq aHYoGycqGH8aapcaaIXaGaai4oaiabeo7aNjabg6da+iaaicdacaGG 7aGaam4yaiabg6da+iaaicdaaaa@4BD6@                          (3.2)

When c=1, the three parameter discrete Burr type XII distribution becomes discrete Lomax distribution with pdf and cdf given by

P( x )= β log( 1+( x/γ ) ) β log( 1+( ( x+1 )/γ ) )    x=0,1,2,3. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGqbWaaeWaa8aabaWdbiaabIhaaiaawIcacaGLPaaacqGH 9aqpcaqGYoWdamaaCaaabeqcfasaa8qaciGGSbGaai4BaiaacEgaju aGdaqadaqcfaYdaeaapeGaaGymaiabgUcaRKqbaoaabmaajuaipaqa a8qacaqG4bGaai4laiaabo7aaiaawIcacaGLPaaaaiaawIcacaGLPa aaaaqcfaOaeyOeI0IaaeOSd8aadaahaaqabKqbGeaapeGaciiBaiaa c+gacaGGNbqcfa4aaeWaaKqbG8aabaWdbiaaigdacqGHRaWkjuaGda qadaqcfaYdaeaajuaGpeWaaeWaaKqbG8aabaWdbiaabIhacqGHRaWk caaIXaaacaGLOaGaayzkaaGaai4laiaabo7aaiaawIcacaGLPaaaai aawIcacaGLPaaaaaqcfaOaaeiOaiaabckacaqGGcGaaeiEaiabg2da 9iaaicdacaGGSaGaaGymaiaacYcacaaIYaGaaiilaiaaiodacqGHMa cVcaGGUaaaaa@6959@                                                  (3.3) 

Where β= e k ; 0 < β<1;γ>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGYoGaeyypa0Jaaeyza8aadaahaaqabKqbGeaapeGaeyOe I0Iaae4AaaaajuaGcaGG7aGaaeiOaiaaicdacaqGGcGaeyipaWJaai iOaiabek7aIjabgYda8iaaigdacaGG7aGaeq4SdCMaeyOpa4JaaGim aaaa@4A12@

F( x )= 1 β log(1+(x/y)) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGgbWdamaabmaabaWdbiaadIhaa8aacaGLOaGaayzkaaWd biabg2da9iaabccacaaIXaGaeyOeI0IaeqOSdi2aaWbaaeqajuaiba GaciiBaiaac+gacaGGNbGaaiikaiaaigdacqGHRaWkcaGGOaGaamiE aiaac+cacaWG5bGaaiykaiaacMcaaaaaaa@4942@

   Where β= e k ;0<β<1;σ>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGYoGaeyypa0Jaaeyza8aadaahaaqabKqbGeaapeGaeyOe I0Iaae4AaaaajuaGcaGG7aGaaGimaiabgYda8iaabk7acaqG8aGaae ymaiaabUdacqaHdpWCcaqG+aGaaeimaaaa@45C0@                              (3.4)

The quantile functions for three parameter discrete Burr type XII and discrete Lomax distributions can be obtained by inverting (3.2) and (3.4) respectively.

x φ =[ γ ( e f( φ,β ) 1 ) 1 c 1 ]   MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqG4bWdamaaBaaajuaibaWdbiaabA8aaKqba+aabeaapeGa eyypa0ZaamWaa8aabaWdbiaabo7adaqadaWdaeaapeGaaeyza8aada ahaaqabKqbGeaapeGaaeOzaKqbaoaabmaajuaipaqaa8qacaqGgpGa aiilaiaabk7aaiaawIcacaGLPaaaaaqcfaOaeyOeI0IaaGymaaGaay jkaiaawMcaa8aadaahaaqabKqbGeaajuaGpeWaaSaaaKqbG8aabaWd biaaigdaa8aabaWdbiaabogaaaaaaKqbakabgkHiTiaaigdaaiaawU facaGLDbaacaqGGcGaaeiOaaaa@51A3@ for DBD-XII and x φ =[ γ( e f( φ,β ) 1 )1 ]     MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqG4bWdamaaBaaajuaibaWdbiaabA8aaKqba+aabeaapeGa eyypa0ZaamWaa8aabaWdbiaabo7adaqadaWdaeaapeGaaeyza8aada ahaaqabKqbGeaapeGaaeOzaKqbaoaabmaajuaipaqaa8qacaqGgpGa aiilaiaabk7aaiaawIcacaGLPaaaaaqcfaOaeyOeI0IaaGymaaGaay jkaiaawMcaaiabgkHiTiaaigdaaiaawUfacaGLDbaacaqGGcGaaeiO aiaabckacaqGGcaaaa@5041@ for DLomax distribution.

Where f( φ,β )= log( 1φ ) logβ  ;c>0, γ>0, β>0  MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGMbWaaeWaa8aabaWdbiaabA8acaGGSaGaaeOSdaGaayjk aiaawMcaaiabg2da9maalaaapaqaa8qacaqGSbGaae4BaiaabEgada qadaWdaeaapeGaaGymaiabgkHiTiaabA8aaiaawIcacaGLPaaaa8aa baWdbiaabYgacaqGVbGaae4zaiaabk7aaaGaaeiOaiaacUdacaqGJb GaeyOpa4JaaGimaiaacYcacaGGGcGaae4Sdiabg6da+iaaicdacaGG SaGaaiiOaiabek7aIjabg6da+iaaicdacaGGGcaaaa@58DC@

Where [ ] denotes the greatest integer function (the largest integer less than or equal). In particular, the median can be written as x 0.5 =[ γ ( e f( β ) 1 ) 1 c 1 ]    MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqG4bWdamaaBaaajuaibaWdbiaaicdacaGGUaGaaGynaaqc fa4daeqaa8qacqGH9aqpdaWadaWdaeaapeGaae4Sdmaabmaapaqaa8 qacaqGLbWdamaaCaaabeqcfasaa8qacaqGMbqcfa4aaeWaaKqbG8aa baWdbiaabk7aaiaawIcacaGLPaaaaaqcfaOaeyOeI0IaaGymaaGaay jkaiaawMcaa8aadaahaaqabKqbGeaajuaGpeWaaSaaaKqbG8aabaWd biaaigdaa8aabaWdbiaabogaaaaaaKqbakabgkHiTiaaigdaaiaawU facaGLDbaacaqGGcGaaeiOaiaabckaaaa@51A9@ for three parameter discrete Burr type XII distribution and for discrete Lomax distribution the median is x 0.5 =[ γ( e f( β ) 1 )1 ]      MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqG4bWdamaaBaaajuaibaWdbiaaicdacaGGUaGaaGynaaqc fa4daeqaa8qacqGH9aqpdaWadaWdaeaapeGaae4Sdmaabmaapaqaa8 qacaqGLbWdamaaCaaabeqcfasaa8qacaqGMbqcfa4aaeWaaKqbG8aa baWdbiaabk7aaiaawIcacaGLPaaaaaqcfaOaeyOeI0IaaGymaaGaay jkaiaawMcaaiabgkHiTiaaigdaaiaawUfacaGLDbaacaqGGcGaaeiO aiaabckacaqGGcGaaeiOaaaa@5047@ Where f( β )= log( 2 ) logβ  ;c>0, γ>0, β>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGMbWaaeWaa8aabaWdbiaabk7aaiaawIcacaGLPaaacqGH 9aqpdaWcaaWdaeaapeGaeyOeI0IaaeiBaiaab+gacaqGNbWaaeWaa8 aabaWdbiaaikdaaiaawIcacaGLPaaaa8aabaWdbiaabYgacaqGVbGa ae4zaiaabk7aaaGaaeiOaiaacUdacaqGJbGaeyOpa4JaaGimaiaacY cacaGGGcGaae4Sdiabg6da+iaaicdacaGGSaGaaiiOaiabek7aIjab g6da+iaaicdaaaa@5471@

The parameter β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHYoGyaaa@3846@  completely determines the pmf (3.1) at x = 0 and = 1. It should be also noted that in this case the p(x) is always monotonic decreasing for x = 1,2,3,4,….

    When c< log( e ( β ) 1 ) log2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGJbGaeyipaWZaaSaaa8aabaWdbiaabYgacaqGVbGaae4z amaabmaapaqaa8qacaqGLbWdamaaCaaabeqcfasaa8qacqaHfiIXju aGdaqadaqcfaYdaeaapeGaaeOSdaGaayjkaiaawMcaaaaajuaGcqGH sislcaaIXaaacaGLOaGaayzkaaaapaqaa8qacaqGSbGaae4BaiaabE gacaaIYaaaaaaa@4977@  Where ( β )= log( 2 β log( 2 ) 1 ) logβ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHfiIXdaqadaWdaeaapeGaaeOSdaGaayjkaiaawMcaaiab g2da9maalaaapaqaa8qacaqGSbGaae4BaiaabEgadaqadaWdaeaape GaaGOmaiaabk7apaWaaWbaaeqajuaibaWdbiGacYgacaGGVbGaai4z aKqbaoaabmaajuaipaqaa8qacaaIYaaacaGLOaGaayzkaaaaaKqbak abgkHiTiaaigdaaiaawIcacaGLPaaaa8aabaWdbiaabYgacaqGVbGa ae4zaiaabk7aaaaaaa@4F4F@

Where β= e k ;0<β<1;c>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHYoGycqGH9aqpcaWGLbWaaWbaaeqajuaibaGaeyOeI0Ia am4AaaaajuaGcaGG7aGaaGimaiabgYda8iabek7aIjabgYda8iaaig dacaGG7aGaam4yaiabg6da+iaaicdaaaa@4637@  otherwise it is no longer monotonic decreasing but is unimodal, having a mode at x=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG4bGaeyypa0JaaGymaaaa@3963@ i.e., it takes a jump at x=1 and then decreases for all x1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG4bGaeyyzImRaaGymaaaa@3A23@ . Figure 5-10 exhibit a graphical overview of the pmf plot for both three parameter discrete Burr type XII and discrete Lomax models for different values of parameters.

Figure 5: pmf plot for DBD-XII (β, c, γ).

Figure 6: pmf plot for DBD-XII (β, c, γ).

Figure 7: pmf plot for DBD-XII (β, c, γ).

Figure 8: pmf plot for DBD-XII (β, c, γ).

Figure 9: pmf plot for DLomax (β, γ).

Figure 10: pmf plot for DLomax (β, γ).

In addition, the modal value of three parameter discrete Burr type XII distribution, x m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaamyBaaqcfayabaaaaa@3950@  is given by x m =[ γ ( c1 clog( β )+1 ) 1 c ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqG4bWdamaaBaaajuaibaWdbiaab2gaaKqba+aabeaapeGa eyypa0ZaamWaa8aabaWdbiaabo7adaqadaWdaeaapeWaaSaaa8aaba WdbiaabogacqGHsislcaaIXaaapaqaa8qacqGHsislcaqGJbGaaeiB aiaab+gacaqGNbWaaeWaa8aabaWdbiaabk7aaiaawIcacaGLPaaacq GHRaWkcaaIXaaaaaGaayjkaiaawMcaa8aadaahaaqabKqbGeaajuaG peWaaSaaaKqbG8aabaWdbiaaigdaa8aabaWdbiaabogaaaaaaaqcfa Oaay5waiaaw2faaaaa@4F42@ , in case when c>1 [if c1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGJbGaeyizImQaaGymaaaa@39FB@ , then the distribution is monotonic decreasing for all x=0,1,2,…..] , the value of c plays a very important role in determining the shape of the cdf curve , the lower the value of c , the sharper the fall of cdf curve, while lower the value of k parameter, the sharper the initial rise of the cdf curve.

When γ1 and c>1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGZoGaeyiyIKRaaGymaiaacckacaWGHbGaamOBaiaadsga caGGGcGaam4yaiabg6da+iaaigdaaaa@4215@ , the distribution of three parameter discrete Burr type XII model can attain model value other than at x=1 and x=0 also. Figure 11-13 provides display of pmf plot when the model value of the distribution is other than at x=1 also.

Figure 11: pmf plot for DBD-XII (β, c, γ).

Figure 12: pmf plot for DBD-XII (β, c, γ).

Figure 13: pmf plot for DBD-XII (β, c, γ).

Reliability Measures of Three Parameter Discrete Burr Type XII Random Variable Dx are given by

  1. Survival function

s( x )=p( dXx )= β log( 1+ ( x/γ ) c ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGZbWaaeWaa8aabaWdbiaabIhaaiaawIcacaGLPaaacqGH 9aqpcaqGWbWaaeWaa8aabaWdbiaabsgacaqGybGaeyyzImRaaeiEaa GaayjkaiaawMcaaiabg2da9iaabk7apaWaaWbaaeqajuaibaWdbiGa cYgacaGGVbGaai4zaKqbaoaabmaajuaipaqaa8qacaaIXaGaey4kaS scfa4aaeWaaKqbG8aabaWdbiaabIhacaGGVaGaae4SdaGaayjkaiaa wMcaaKqba+aadaahaaqcfasabeaapeGaae4yaaaaaiaawIcacaGLPa aaaaaaaa@530E@ where β= e k ; 0 <β<1;γ>0;c>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGYoGaeyypa0Jaaeyza8aadaahaaqabKqbGeaapeGaeyOe I0Iaae4AaaaajuaGcaGG7aGaaeiOaiaaicdacaqGGcGaaeipaiabek 7aIjaabYdacaqGXaGaae4oaiabeo7aNjaab6dacaqGWaGaae4oaiaa bogacaqG+aGaaeimaaaa@4B26@ x=0,1,2, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqG4bGaeyypa0JaaGimaiaacYcacaaIXaGaaiilaiaaikda caGGSaGaeyOjGWlaaa@3E75@

s( x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGZbWaaeWaa8aabaWdbiaabIhaaiaawIcacaGLPaaaaaa@3A3E@ is same for continuous Burr type XII distribution and discrete Burr type XII distribution at the integer points of x.

  1. Rate of Failure, r(x) is given by

r( x )= p( x ) s( x ) = β log( 1+( x/γ ) ) β log( 1+( ( x+1 )/γ ) ) β log( 1+ ( x/γ ) c ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGYbWaaeWaa8aabaWdbiaabIhaaiaawIcacaGLPaaacqGH 9aqpdaWcaaWdaeaapeGaaeiCamaabmaapaqaa8qacaqG4baacaGLOa Gaayzkaaaapaqaa8qacaqGZbWaaeWaa8aabaWdbiaabIhaaiaawIca caGLPaaaaaGaeyypa0ZaaSaaa8aabaWdbiaabk7apaWaaWbaaeqaju aibaWdbiGacYgacaGGVbGaai4zaKqbaoaabmaajuaipaqaa8qacaaI XaGaey4kaSscfa4aaeWaaKqbG8aabaWdbiaabIhacaGGVaGaae4Sda GaayjkaiaawMcaaaGaayjkaiaawMcaaaaajuaGcqGHsislcaqGYoWd amaaCaaabeqcfasaa8qaciGGSbGaai4BaiaacEgajuaGdaqadaqcfa YdaeaapeGaaGymaiabgUcaRKqbaoaabmaajuaipaqaaKqba+qadaqa daqcfaYdaeaapeGaaeiEaiabgUcaRiaaigdaaiaawIcacaGLPaaaca GGVaGaae4SdaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaaaKqba+aa baWdbiaabk7apaWaaWbaaeqajuaibaWdbiGacYgacaGGVbGaai4zaK qbaoaabmaajuaipaqaa8qacaaIXaGaey4kaSscfa4aaeWaaKqbG8aa baWdbiaabIhacaGGVaGaae4SdaGaayjkaiaawMcaaKqba+aadaahaa qcfasabeaapeGaae4yaaaaaiaawIcacaGLPaaaaaaaaaaa@754A@

where β= e k ; 0 <β<1;γ>0;c>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGYoGaeyypa0Jaaeyza8aadaahaaqabKqbGeaapeGaeyOe I0Iaae4AaaaajuaGcaGG7aGaaeiOaiaaicdacaqGGcGaaeipaiabek 7aIjaabYdacaqGXaGaae4oaiabeo7aNjaab6dacaqGWaGaae4oaiaa bogacaqG+aGaaeimaaaa@4B26@     x=0,1,2,

MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqG4bGaeyypa0JaaGimaiaacYcacaaIXaGaaiilaiaaikda caGGSaGaeyOjGWlaaa@3E75@

  1. Second Rate of Failure is given by

SRF( x )=log( β log( 1+ ( x/γ ) c ) β log( 1+ ( ( x+1 )/γ ) c ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGtbGaaeOuaiaabAeadaqadaWdaeaapeGaaeiEaaGaayjk aiaawMcaaiabg2da9iaabYgacaqGVbGaae4zamaabmaapaqaa8qada WcaaWdaeaapeGaaeOSd8aadaahaaqabKqbGeaapeGaciiBaiaac+ga caGGNbqcfa4aaeWaaKqbG8aabaWdbiaaigdacqGHRaWkjuaGdaqada qcfaYdaeaapeGaaeiEaiaac+cacaqGZoaacaGLOaGaayzkaaqcfa4d amaaCaaajuaibeqaa8qacaqGJbaaaaGaayjkaiaawMcaaaaaaKqba+ aabaWdbiaabk7apaWaaWbaaeqajuaibaWdbiGacYgacaGGVbGaai4z aKqbaoaabmaajuaipaqaa8qacaaIXaGaey4kaSscfa4aaeWaaKqbG8 aabaqcfa4dbmaabmaajuaipaqaa8qacaqG4bGaey4kaSIaaGymaaGa ayjkaiaawMcaaiaac+cacaqGZoaacaGLOaGaayzkaaqcfa4damaaCa aajuaibeqaa8qacaqGJbaaaaGaayjkaiaawMcaaaaaaaaajuaGcaGL OaGaayzkaaaaaa@65EB@ where β= e k ; 0 <β<1;γ>0;c>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGYoGaeyypa0Jaaeyza8aadaahaaqabKqbGeaapeGaeyOe I0Iaae4AaaaajuaGcaGG7aGaaeiOaiaaicdacaqGGcGaaeipaiabek 7aIjaabYdacaqGXaGaae4oaiabeo7aNjaab6dacaqGWaGaae4oaiaa bogacaqG+aGaaeimaaaa@4B26@     x=0,1,2, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqG4bGaeyypa0JaaGimaiaacYcacaaIXaGaaiilaiaaikda caGGSaGaeyOjGWlaaa@3E75@

The reliability measures for discrete Lomax distribution can be directly obtained from reliability measures of three parameter discrete Burr type XII distribution by taking c=1.

It could be seen that r(x) and SRF(x) are always monotonic decreasing functions if

γ=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGZoGaeyypa0JaaGymaaaa@399F@

and c<log[ e ϕ(β) 1]/log2=α(say) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGJbGaeyipaWJaamiBaiaad+gacaWGNbGaai4waiaadwga paWaaWbaaeqajuaibaWdbiabew9aMjaacIcacqaHYoGycaGGPaaaaK qbakabgkHiTiaaigdacaGGDbGaai4laiaadYgacaWGVbGaam4zaiaa ikdacqGH9aqpcqaHXoqycaGGOaGaam4CaiaadggacaWG5bGaaiykaa aa@508D@ Where ( β )= log( β 2log2 ) logβ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHfiIXdaqadaWdaeaapeGaaeOSdaGaayjkaiaawMcaaiab g2da9maalaaapaqaa8qacaqGSbGaae4BaiaabEgadaqadaWdaeaape GaaeOSd8aadaahaaqabKqbGeaapeGaaGOmaiaabYgacaqGVbGaae4z aiaaikdaaaaajuaGcaGLOaGaayzkaaaapaqaa8qacaqGSbGaae4Bai aabEgacaqGYoaaaaaa@4B3E@

β= e (k) ;0<β<1;σ>0;c>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHYoGycqGH9aqpcaWGLbWdamaaCaaabeqcfasaa8qacaGG OaGaeyOeI0Iaam4AaiaacMcaaaqcfaOaai4oaiaaicdacqGH8aapcq aHYoGycqGH8aapcaaIXaGaai4oaiabeo8aZjabg6da+iaaicdacaGG 7aGaam4yaiabg6da+iaaicdaaaa@4BF3@

Figure 14-19 illustrates the second rate of failure plot for DBD-XII and discrete Lomax models for different values of parameters. For c >α; r(0)< r(1) and SRF(0)< SRF(1) and for all other values of x ≥ 1, r(x) and SRF(x) decreases, clearly the hazard rates of continuous model and the discrete modal shows the same monotonocity. In case γ1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGZoGaeyiyIKRaaGymaaaa@3A5F@ the hazard rate function for three parameter Burr type XII can attain maximum at other than x=0 and x=1 also as illustrated in Figure 18.

Moments of three parameter discrete burr type XII distribution and discrete lomax distribution

Figure 14: SRF(x) plot for DBD-XII (β, c, γ).

Figure 15: SRF(x) plot for DBD-XII (β, c, γ).

Figure 16: SRF(x) plot for DBD-XII (β, c, γ).

Figure 17: SRF(x) plot for DBD-XII (β, c, γ).

Figure 18: SRF(x) plot for DBD-XII (β, c, γ).

Figure 19: SRF(x) plot for DBD-XII (β, c, γ).

E( x r )= x=0 x r p( x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGfbWaaeWaa8aabaWdbiaabIhapaWaaWbaaeqajuaibaWd biaabkhaaaaajuaGcaGLOaGaayzkaaGaeyypa0ZaaybCaeqajuaipa qaa8qacaqG4bGaeyypa0JaaGimaaWdaeaapeGaeqOhIukajuaGpaqa a8qacqGHris5aaGaaeiEa8aadaahaaqabKqbGeaapeGaaeOCaaaaju aGcaqGWbWaaeWaa8aabaWdbiaabIhaaiaawIcacaGLPaaaaaa@4B00@

= x=1 [ x r ( x1 ) r ]s( x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGH9aqpdaGfWbqabKqbG8aabaWdbiaabIhacqGH9aqpcaaI Xaaapaqaa8qacqaHEisPaKqba+aabaWdbiabggHiLdaacaGGBbGaae iEa8aadaahaaqabKqbGeaapeGaaeOCaaaajuaGcqGHsisldaqcWaWd aeaapeGaaeiEaiabgkHiTiaaigdacaGGPaWdamaaCaaabeqcfasaa8 qacaqGYbaaaaqcfaOaayjkaiaaw2faaiaabohadaqadaWdaeaapeGa aeiEaaGaayjkaiaawMcaaaaa@4EC6@

Now, E( x )= 1 s( x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGfbWaaeWaa8aabaWdbiaabIhaaiaawIcacaGLPaaacqGH 9aqpdaGfWbqabKqbG8aabaWdbiaaigdaa8aabaWdbiabe6HiLcqcfa 4daeaapeGaeyyeIuoaaiaabohadaqadaWdaeaapeGaaeiEaaGaayjk aiaawMcaaaaa@4424@ = 1 β log( 1+ ( x/γ ) c ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGH9aqpdaGfWbqabKqbG8aabaWdbiaaigdaa8aabaWdbiab e6HiLcqcfa4daeaapeGaeyyeIuoaaiaabk7apaWaaWbaaeqajuaiba WdbiGacYgacaGGVbGaai4zaKqbaoaabmaajuaipaqaa8qacaaIXaGa ey4kaSscfa4aaeWaaKqbG8aabaWdbiaabIhacaGGVaGaae4SdaGaay jkaiaawMcaaKqba+aadaahaaqcfasabeaapeGaae4yaaaaaiaawIca caGLPaaaaaaaaa@4CC5@

E( x 2 )= 1 ( 2x1 )s( x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGfbWaaeWaa8aabaWdbiaabIhapaWaaWbaaeqajuaibaWd biaaikdaaaaajuaGcaGLOaGaayzkaaGaeyypa0ZaaybCaeqajuaipa qaa8qacaaIXaaapaqaa8qacqaHEisPaKqba+aabaWdbiabggHiLdaa daqadaWdaeaapeGaaGOmaiaabIhacqGHsislcaaIXaaacaGLOaGaay zkaaGaae4Camaabmaapaqaa8qacaqG4baacaGLOaGaayzkaaaaaa@4AE4@ = 1 ( 2x1 ) β log( 1+ ( x/γ ) c ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGH9aqpdaGfWbqabKqbG8aabaWdbiaaigdaa8aabaWdbiab e6HiLcqcfa4daeaapeGaeyyeIuoaamaabmaapaqaa8qacaaIYaGaae iEaiabgkHiTiaaigdaaiaawIcacaGLPaaacaqGYoWdamaaCaaabeqc fasaa8qaciGGSbGaai4BaiaacEgajuaGdaqadaqcfaYdaeaapeGaaG ymaiabgUcaRKqbaoaabmaajuaipaqaa8qacaqG4bGaai4laiaabo7a aiaawIcacaGLPaaajuaGpaWaaWbaaKqbGeqabaWdbiaabogaaaaaca GLOaGaayzkaaaaaaaa@51CD@

V( x )= 1 ( 2x1 ) β log( 1+ ( x/γ ) c ) { 1 ( 2x1 ) β log( 1+ ( x/γ ) c ) } 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGwbWdamaabmaabaWdbiaadIhaa8aacaGLOaGaayzkaaGa eyypa0ZdbmaawahabeqcfaYdaeaapeGaaGymaaWdaeaapeGaeqOhIu kajuaGpaqaa8qacqGHris5aaWaaeWaa8aabaWdbiaaikdacaqG4bGa eyOeI0IaaGymaaGaayjkaiaawMcaaiaabk7apaWaaWbaaeqajuaiba WdbiGacYgacaGGVbGaai4zaKqbaoaabmaajuaipaqaa8qacaaIXaGa ey4kaSscfa4aaeWaaKqbG8aabaWdbiaabIhacaGGVaGaae4SdaGaay jkaiaawMcaaKqba+aadaahaaqcfasabeaapeGaae4yaaaaaiaawIca caGLPaaaaaqcfaOaeyOeI0YaaiWaa8aabaWdbmaawahabeqcfaYdae aapeGaaGymaaWdaeaapeGaeqOhIukajuaGpaqaa8qacqGHris5aaWa aeWaa8aabaWdbiaaikdacaqG4bGaeyOeI0IaaGymaaGaayjkaiaawM caaiaabk7apaWaaWbaaeqajuaibaWdbiGacYgacaGGVbGaai4zaKqb aoaabmaajuaipaqaa8qacaaIXaGaey4kaSscfa4aaeWaaKqbG8aaba WdbiaabIhacaGGVaGaae4SdaGaayjkaiaawMcaaKqba+aadaahaaqc fasabeaapeGaae4yaaaaaiaawIcacaGLPaaaaaaajuaGcaGL7bGaay zFaaWdamaaCaaabeqcfasaa8qacaaIYaaaaaaa@7511@

For checking purpose of moments convergence or divergence, we have

E( x r )= x=1 [ x r (x1) r ]s(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGfbGaaiikaiaadIhapaWaaWbaaeqajuaibaWdbiaadkha aaqcfaOaaiykaiabg2da9maaqadabaGaai4waiaadIhapaWaaWbaae qajuaibaWdbiaadkhaaaqcfaOaeyOeI0IaaiikaiaadIhacqGHsisl caaIXaGaaiyka8aadaahaaqabKqbGeaapeGaamOCaaaajuaGcaGGDb Gaam4CaiaacIcacaWG4bGaaiykaaqcfasaaiaadIhacqGH9aqpcaaI XaaabaGaeyOhIukajuaGcqGHris5aaaa@5271@     r x=1 x r1 β log( 1+ ( x/γ ) c ) ck x=1 1 x ckr+1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGHKjYOcaqGYbWaaybCaeqajuaipaqaa8qacaqG4bGaeyyp a0JaaGymaaWdaeaapeGaeqOhIukajuaGpaqaa8qacqGHris5aaGaae iEa8aadaahaaqabKqbGeaapeGaaeOCaiabgkHiTiaaigdaaaqcfaOa aeOSd8aadaahaaqabKqbGeaapeGaciiBaiaac+gacaGGNbqcfa4aae WaaKqbG8aabaWdbiaaigdacqGHRaWkjuaGdaqadaqcfaYdaeaapeGa aeiEaiaac+cacaqGZoaacaGLOaGaayzkaaqcfa4damaaCaaajuaibe qaa8qacaqGJbaaaaGaayjkaiaawMcaaaaajuaGcqGHKjYOcaqGYbGa ae4Sd8aadaahaaqabKqbGeaapeGaae4yaiaabUgaaaqcfa4aaybCae qajuaipaqaa8qacaqG4bGaeyypa0JaaGymaaWdaeaapeGaeqOhIuka juaGpaqaa8qacqGHris5aaWaaSaaa8aabaWdbiaaigdaa8aabaWdbi aabIhapaWaaWbaaeqajuaibaWdbiaabogacaqGRbGaeyOeI0IaaeOC aiabgUcaRiaaigdaaaaaaaaa@6B7E@

Where β= e k ; 0 <β<1;γ>0;c>0;k=lo g e β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGYoGaeyypa0Jaaeyza8aadaahaaqabKqbGeaapeGaeyOe I0Iaae4AaaaajuaGcaGG7aGaaeiOaiaaicdacaqGGcGaaeipaiabek 7aIjaabYdacaqGXaGaae4oaiabeo7aNjaab6dacaqGWaGaae4oaiaa bogacaqG+aGaaGimaiaacUdacaWGRbGaeyypa0JaeyOeI0IaamiBai aad+gacaWGNbWdamaaBaaajuaibaWdbiaadwgaaKqba+aabeaapeGa eqOSdigaaa@5545@ which is convergent if ck-r+1>1 or ck>r

In case of discrete Lomax distribution, for the convergence of moments k should be greater than r. Hence, E( x r ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGfbWaaeWaa8aabaWdbiaabIhapaWaaWbaaeqajuaibaWd biaabkhaaaaajuaGcaGLOaGaayzkaaaaaa@3C02@ for three parameter Burr type XII distribution and discrete Lomax distribution exists if and only if ck>r and k>r respectively. Or in other words when β< e r/c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHYoGycqGH8aapcaWGLbWdamaaCaaabeqcfasaa8qacqGH sislcaWGYbGaai4laiaadogaaaaaaa@3E22@ moments of three parameter Burr type XII distribution exists. There is a one to one correspondence between three parameter continuous Burr type XII distribution and three parameter discrete Burr type XII distribution, as the expressions for survival function, failure rate function, second rate of failure function for DBD-XII ( β,c,γ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aaqaaaaaaaaaWdbiaabk7acaGGSaGaae4yaiaacYcacaqGZoaapaGa ayjkaiaawMcaaaaa@3CF4@ can be directly obtained from continuous Burr type XII distribution by replacing k= log e β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGRbGaeyypa0JaeyOeI0IaaeiBaiaab+gacaqGNbWdamaa BaaajuaibaWdbiaabwgaaKqba+aabeaapeGaaeOSdaaa@3F8C@ .

Table 1 and Table 2 exhibits the index of dispersion D = [E(X2) − (E(X))2]/E(X), for different values of the parameters c, β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGYoaaaa@37DD@ and γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGZoaaaa@37DE@ for three parameter discrete Burr type XII distribution and discrete Lomax distribution. It can be seen that this variance to mean ratio goes on increasing in case of discrete Lomax distribution as the parameters goes on increasing, and therefore in this case the discrete Lomax distribution seems over dispersed. In case of discrete Burr type XII as β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGYoaaaa@37DD@ and c goes on increasing the distribution shows under dispersion.

Different Values of γ

Different Values of β

Parameters

0.0001

0.0003

0.0009

0.0060

0.0200

0.0300

0.0400

0.0500

0.10

1.0060

1.0120

1.0250

1.1030

1.3000

1.4610

1.6510

1.8850

0.11

1.0060

1.0120

1.0260

1.1060

1.3060

1.4690

1.6630

1.8990

0.12

1.0060

1.0120

1.0260

1.1080

1.3120

1.4780

1.6740

1.9140

0.14

1.0060

1.0130

1.0280

1.1130

1.3240

1.4950

1.6960

1.9430

0.17

1.0070

1.0150

1.0310

1.1210

1.3420

1.5210

1.7310

1.9860

0.20

1.0080

1.0160

1.0330

1.1300

1.3610

1.5470

1.7650

2.0300

0.25

1.0100

1.0190

1.0380

1.1440

1.3930

1.5900

1.8220

2.1030

0.33

1.0120

1.0240

1.0470

1.1680

1.4440

1.6610

1.9140

2.2200

0.50

1.0200

1.0360

1.0680

1.2220

1.5550

1.8120

2.1090

2.4680

1.00

1.0530

1.0870

1.1450

1.3940

1.8850

2.2550

2.6800

3.1940

1.11

1.0620

1.1000

1.1640

1.4330

1.9580

2.3520

2.8060

3.3530

2.00

1.1530

1.2220

1.3310

1.7550

2.5520

3.1450

3.8280

4.6530

2.50

1.2110

1.2970

1.4310

1.9400

2.8890

3.5940

4.4080

5.3900

3.33

1.3140

1.4270

1.6020

2.2520

3.4540

4.3460

5.3760

6.6210

5.00

1.5380

1.7060

1.9600

2.8920

4.6040

5.8730

7.3390

9.1120

10.00

2.2650

2.5920

3.0810

4.8530

8.0870

10.4860

13.2590

16.6180

Table 1: Index of dispersion for DLomax for different values of β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@ and γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC gaaa@382B@ .

Different Values of c

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

Parameters

0.0100

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

2

0.9900

0.9945

1.0035

1.0156

1.0307

1.0484

1.0688

1.092

1.1181

3

0.9609

0.9391

0.9222

0.9083

0.8967

0.887

0.8789

0.8722

0.8668

4

0.9590

0.934

0.9131

0.8945

0.8778

0.8624

0.8481

0.8349

0.8225

5

0.9589

0.9336

0.9121

0.8928

0.8751

0.8584

0.8428

0.8279

0.8137

6

0.9589

0.9336

0.912

0.8926

0.8747

0.8578

0.8419

0.8266

0.812

7

0.9589

0.9336

0.912

0.8926

0.8746

0.8578

0.8417

0.8264

0.8117

Different Values of c

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

Parameters

0.1100

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

2

1.1803

1.2169

1.2579

1.3036

1.3548

1.4122

1.4768

1.5497

1.6324

3

0.8598

0.8582

0.8577

0.8584

0.8604

0.8636

0.8682

0.8742

0.8816

4

0.8003

0.7904

0.7811

0.7726

0.7647

0.7576

0.7511

0.7453

0.7403

5

0.7872

0.7748

0.7628

0.7514

0.7404

0.7299

0.7198

0.7102

0.701

6

0.7843

0.7711

0.7583

0.746

0.7339

0.7222

0.7109

0.6998

0.6891

7

0.7836

0.7703

0.7572

0.7445

0.7322

0.7201

0.7083

0.6967

0.6854

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

Different Values of c

Parameters

0.4100

0.42

0.43

0.5

0.51

0.52

0.53

0.54

0.6

5

0.654

0.6636

0.6756

0.848

0.8929

0.9461

1.0094

1.0854

1.2897

6

0.5500

0.5504

0.5518

0.6015

0.617

0.6356

0.6578

0.6844

0.7539

7

0.5046

0.5007

0.4973

0.4969

0.5013

0.5074

0.5153

0.5253

0.5531

8

0.4831

0.4769

0.4711

0.4454

0.4446

0.4447

0.4459

0.4483

0.4577

9

0.4724

0.465

0.4579

0.4181

0.4143

0.4112

0.4088

0.4071

0.4068

10

0.4670

0.4589

0.451

0.4029

0.3974

0.3923

0.3878

0.3838

0.3777

Table 2: Index of dispersion for DBD-XII for different values of β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@  and c when γ=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC Maeyypa0JaaGymaaaa@39EC@ .

Estimation of the Parameters of Three Parameter Discrete Burr Type XII Distribution and Discrete Lomax Distribution

Estimation of the parameters based on the ML method: Let X 1, X 2 X 3, X n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGybWdamaaBaaajuaibaWdbiaaigdacaGGSaaajuaGpaqa baWdbiaabIfapaWaaSbaaKqbGeaapeGaaGOmaaqcfa4daeqaa8qaca qGybWdamaaBaaajuaibaWdbiaaiodacaGGSaaajuaGpaqabaWdbiab gAci8kabgAci8kaabIfapaWaaSbaaKqbGeaapeGaaeOBaaqcfa4dae qaaaaa@460E@ be a random sample of size n. If these X i. 's MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGybWdamaaBaaajuaibaWdbiaabMgacaGGUaaajuaGpaqa baWdbiaabEcacaqGZbaaaa@3BD9@ are assumed to be iid random variables following three parameter discrete Burr type XII distribution i.e., DBDXII( β,c,γ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGebGaaeOqaiaabseacqGHsislcaqGybGaaeysaiaabMea daqadaWdaeaapeGaaeOSdiaacYcacaqGJbGaaiilaiaabo7aaiaawI cacaGLPaaaaaa@42B7@ their likelihood function is given by L(β,c,γ;x)= i=1 n p( x i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGmbGaaiikaiabek7aIjaacYcacaWGJbGaaiilaiabeo7a NjaacUdacaWG4bGaaiykaiabg2da9maaradabaGaamiCaiaacIcaca GG4bWaaSbaaKqbGeaacaWGPbaajuaGbeaacaGGPaaajuaibaGaamyA aiabg2da9iaaigdaaeaacaWGUbaajuaGcqGHpis1aaaa@4C6C@

= i=1 n ( β log( 1+ ( x/γ ) c ) β log( 1+ ( ( x+1 )/γ ) c ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGH9aqpdaGfWbqabKqbG8aabaWdbiaabMgacqGH9aqpcaaI Xaaapaqaa8qacaqGUbaajuaGpaqaa8qacqGHpis1aaWaaeWaa8aaba Wdbiaabk7apaWaaWbaaeqajuaibaWdbiGacYgacaGGVbGaai4zaKqb aoaabmaajuaipaqaa8qacaaIXaGaey4kaSscfa4aaeWaaKqbG8aaba WdbiaabIhacaGGVaGaae4SdaGaayjkaiaawMcaaKqba+aadaahaaqc fasabeaapeGaae4yaaaaaiaawIcacaGLPaaaaaqcfaOaeyOeI0Iaae OSd8aadaahaaqabKqbGeaapeGaciiBaiaac+gacaGGNbqcfa4aaeWa aKqbG8aabaWdbiaaigdacqGHRaWkjuaGdaqadaqcfaYdaeaajuaGpe WaaeWaaKqbG8aabaWdbiaabIhacqGHRaWkcaaIXaaacaGLOaGaayzk aaGaai4laiaabo7aaiaawIcacaGLPaaajuaGpaWaaWbaaKqbGeqaba WdbiaabogaaaaacaGLOaGaayzkaaaaaaqcfaOaayjkaiaawMcaaaaa @6580@ (5.1)

And (5.1) can be rewritten as follows L( β,c,γ;x )= i=1 n β log( 1+ ( x/γ ) c ) ( 1 β ( x i ,c,γ ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGmbWaaeWaa8aabaWdbiaabk7acaGGSaGaae4yaiaacYca caqGZoGaai4oaiaabIhaaiaawIcacaGLPaaacqGH9aqpdaGfWbqabK qbG8aabaWdbiaabMgacqGH9aqpcaaIXaaapaqaa8qacaqGUbaajuaG paqaa8qacqGHpis1aaGaaeOSd8aadaahaaqabKqbGeaapeGaciiBai aac+gacaGGNbqcfa4aaeWaaKqbG8aabaWdbiaaigdacqGHRaWkjuaG daqadaqcfaYdaeaapeGaaeiEaiaac+cacaqGZoaacaGLOaGaayzkaa qcfa4damaaCaaajuaibeqaa8qacaqGJbaaaaGaayjkaiaawMcaaaaa juaGdaqadaWdaeaapeGaaGymaiabgkHiTiaabk7apaWaaWbaaeqaju aibaWdbiabewGigNqbaoaabmaajuaipaqaa8qacaqG4bqcfa4damaa BaaajuaibaWdbiaabMgaa8aabeaapeGaaiilaiaabogacaGGSaGaae 4SdaGaayjkaiaawMcaaaaaaKqbakaawIcacaGLPaaaaaa@677F@ (5.2)

where ( x i ,c,γ )=log[ ( 1+ ( ( x i +1)/γ ) c ) ( 1+ ( x i /γ) c ) ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHfiIXdaqadaWdaeaapeGaaeiEa8aadaWgaaqaa8qacaqG PbaapaqabaWdbiaacYcacaqGJbGaaiilaiaabo7aaiaawIcacaGLPa aacqGH9aqpcaqGSbGaae4BaiaabEgacaGGBbWaaSaaa8aabaWdbmaa bmaapaqaa8qacaaIXaGaey4kaSYaaeWaa8aabaWdbiaacIcacaqG4b WdamaaBaaajuaibaWdbiaabMgaaKqba+aabeaapeGaey4kaSIaaGym aiaacMcacaGGVaGaae4SdaGaayjkaiaawMcaa8aadaahaaqabKqbGe aapeGaae4yaaaaaKqbakaawIcacaGLPaaaa8aabaWdbmaabmaapaqa a8qacaaIXaGaey4kaSIaaiikaiaabIhapaWaaSbaaKqbGeaapeGaae yAaaqcfa4daeqaa8qacaGGVaGaae4SdiaacMcapaWaaWbaaeqajuai baWdbiaabogaaaaajuaGcaGLOaGaayzkaaaaaiaac2faaaa@5F8B@

logL= [ log( 1+ ( x/γ ) c )logβ+log( 1 β ( x i ,c,γ ) ) ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGSbGaae4BaiaabEgacaqGmbGaeyypa0Zdamaavacabeqa beaacaaMb8oabaWdbiabggHiLdaadaWadaWdaeaapeGaciiBaiaac+ gacaGGNbWaaeWaa8aabaWdbiaaigdacqGHRaWkdaqadaWdaeaapeGa aeiEaiaac+cacaqGZoaacaGLOaGaayzkaaWdamaaCaaabeqcfasaa8 qacaqGJbaaaaqcfaOaayjkaiaawMcaaiaabYgacaqGVbGaae4zaiaa bk7acqGHRaWkcaqGSbGaae4BaiaabEgadaqadaWdaeaapeGaaGymai abgkHiTiaabk7apaWaaWbaaeqajuaibaWdbiabewGigNqbaoaabmaa juaipaqaa8qacaqG4bqcfa4damaaBaaajuaibaWdbiaabMgaa8aabe aapeGaaiilaiaabogacaGGSaGaae4SdaGaayjkaiaawMcaaaaaaKqb akaawIcacaGLPaaaaiaawUfacaGLDbaaaaa@64F9@ (5.3)

Taking partial derivatives with respect to β,c and γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGYoGaaiilaiaabogacaqGGcGaaeyyaiaab6gacaqGKbGa aeiOaiaabo7aaaa@3FAE@ and equating them to zero, we obtain the normal equations. Which can be solved to obtain the maximum likelihood estimators. logL β = i=1 n [ log( 1+ ( x i /γ) c ) β ̂ ( x i ,c,γ ) β ̂ ( x i ,c,γ )1 ) 1 β ̂ ( x i ,c,γ ) ) ]=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWcaaWdaeaapeGaeyOaIyRaaeiBaiaab+gacaqGNbGaaeit aaWdaeaapeGaeyOaIyRaaeOSdaaacqGH9aqpdaGfWbqabKqbG8aaba WdbiaabMgacqGH9aqpcaaIXaaapaqaa8qacaqGUbaajuaGpaqaa8qa cqGHris5aaWaamWaa8aabaWdbmaalaaapaqaa8qacaqGSbGaae4Bai aabEgadaqadaWdaeaapeGaaGymaiabgUcaRiaacIcacaqG4bWdamaa BaaajuaibaWdbiaabMgaaKqba+aabeaapeGaai4laiaabo7acaGGPa WdamaaCaaabeqcfasaa8qacaqGJbaaaaqcfaOaayjkaiaawMcaaaWd aeaadaWfGaqaa8qacaqGYoaapaqabeaapeGaeSOadqcaaaaacqGHsi sldaWcaaWdaeaapeGaeqybIy8aaeWaa8aabaWdbiaabIhapaWaaSba aKqbGeaapeGaaeyAaaqcfa4daeqaa8qacaGGSaGaae4yaiaacYcaca qGZoaacaGLOaGaayzkaaWdamaaxacabaWdbiaabk7aa8aabeqaa8qa cqWIcmajaaWdamaaCaaabeqcfasaa8qacqaHfiIXjuaGdaqadaqcfa YdaeaapeGaaeiEaKqba+aadaWgaaqcfasaa8qacaqGPbaapaqabaWd biaacYcacaqGJbGaaiilaiaabo7aaiaawIcacaGLPaaacqGHsislca aIXaaaaKqbakaacMcaa8aabaWdbiaaigdacqGHsislpaWaaCbiaeaa peGaaeOSdaWdaeqabaWdbiablkWaKaaapaWaaWbaaeqajuaibaWdbi abewGigNqbaoaabmaajuaipaqaa8qacaqG4bqcfa4damaaBaaajuai baWdbiaabMgaa8aabeaapeGaaiilaiaabogacaGGSaGaae4SdaGaay jkaiaawMcaaaaajuaGcaGGPaaaaaGaay5waiaaw2faaiabg2da9iaa icdaaaa@863B@ (5.4)

logL c = i=1 n [ (( x i /γ ) c ̂ ) logβlogx i 1+ ( x γ ) c ̂ logβ'( x i , c, ̂ γ ) β ( x i , c ̂ ,γ ) 1 β ( x i , c, ̂ γ ) ]   =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWcaaWdaeaapeGaeyOaIyRaaeiBaiaab+gacaqGNbGaaeit aaWdaeaapeGaeyOaIyRaae4yaaaacqGH9aqpdaGfWbqabKqbG8aaba WdbiaabMgacqGH9aqpcaaIXaaapaqaa8qacaqGUbaajuaGpaqaa8qa cqGHris5aaWaamWaa8aabaWdbmaalaaapaqaa8qacaGGOaWaaeWaa8 aabaWdbiaabIhapaWaaSbaaeaapeGaaeyAaaWdaeqaa8qacaGGVaGa ae4SdiaacMcapaWaaWbaaeqabaWaaCbiaKqbGeaapeGaae4yaaqcfa 4daeqabaWdbiablkWaKaaaaaaacaGLOaGaayzkaaGaaeiBaiaab+ga caqGNbGaaeOSdiaabYgacaqGVbGaae4zaiaabIhapaWaaSbaaKqbGe aapeGaaeyAaaqcfa4daeqaaaqaa8qacaaIXaGaey4kaSYaaeWaa8aa baWdbmaalaaapaqaa8qacaqG4baapaqaa8qacaqGZoaaaaGaayjkai aawMcaa8aadaahaaqabeaadaWfGaqcfasaa8qacaqGJbaajuaGpaqa beaapeGaeSOadqcaaaaaaaGaeyOeI0YaaSaaa8aabaWdbiaabYgaca qGVbGaae4zaiaabk7acqaHfiIXcaqGNaWaaeWaa8aabaWdbiaabIha paWaaSbaaKqbGeaapeGaaeyAaaqcfa4daeqaa8qacaGGSaWdamaaxa cabaWdbiaabogacaGGSaaapaqabeaapeGaeSOadqcaaiaabo7aaiaa wIcacaGLPaaacaqGYoWdamaaCaaabeqcfasaa8qacqaHfiIXjuaGda qadaqcfaYdaeaapeGaaeiEaKqba+aadaWgaaqcfasaa8qacaqGPbaa paqabaWdbiaacYcajuaGpaWaaCbiaKqbGeaapeGaae4yaaWdaeqaba WdbiablkWaKaaacaGGSaGaae4SdaGaayjkaiaawMcaaaaaaKqba+aa baWdbiaaigdacqGHsislcaqGYoWdamaaCaaabeqcfasaa8qacqaHfi IXjuaGdaqadaqcfaYdaeaapeGaaeiEaKqba+aadaWgaaqcfasaa8qa caqGPbaapaqabaWdbiaacYcajuaGpaWaaCbiaKqbGeaapeGaae4yai aacYcaa8aabeqaa8qacqWIcmajaaGaae4SdaGaayjkaiaawMcaaaaa aaaajuaGcaGLBbGaayzxaaGaaeiOaiaabckacaqGGcGaeyypa0JaaG imaaaa@99C1@ (5.5)

Where '( x i , c, ̂ γ )= ( x i ,c,γ ) c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHfiIXcaqGNaWaaeWaa8aabaWdbiaabIhapaWaaSbaaKqb GeaapeGaaeyAaaqcfa4daeqaa8qacaGGSaWdamaaxacabaWdbiaabo gacaGGSaaapaqabeaapeGaeSOadqcaaiaabo7aaiaawIcacaGLPaaa cqGH9aqpdaWcaaWdaeaapeGaeyOaIyRaeqybIy8aaeWaa8aabaWdbi aabIhapaWaaSbaaKqbGeaapeGaaeyAaaqcfa4daeqaa8qacaGGSaGa ae4yaiaacYcacaqGZoaacaGLOaGaayzkaaaapaqaa8qacqGHciITca qGJbaaaaaa@50CB@ (5.6)

logL γ = i=1 n [ ( cx i c γ ̂ ( c+1 ) )logβ 1+ ( x γ ̂ ) c logβ'( x i ,c, γ ̂ ) β ( x i ,c, γ ̂ ) β ( x i ,c, γ ̂ ) 1 ]   =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWcaaWdaeaapeGaeyOaIyRaaeiBaiaab+gacaqGNbGaaeit aaWdaeaapeGaeyOaIyRaae4SdaaacqGH9aqpdaGfWbqabKqbG8aaba WdbiaabMgacqGH9aqpcaaIXaaapaqaa8qacaqGUbaajuaGpaqaa8qa cqGHris5aaWaamWaa8aabaWdbmaalaaapaqaa8qacqGHsisldaqada WdaeaapeGaae4yaiaabIhapaWaaSbaaKqbGeaapeGaaeyAaaqcfa4d aeqaamaaCaaabeqcfasaa8qacaqGJbaaaKqba+aadaWfGaqaa8qaca qGZoaapaqabeaapeGaeSOadqcaa8aadaahaaqabKqbGeaapeGaeyOe I0scfa4aaeWaaKqbG8aabaWdbiaabogacqGHRaWkcaaIXaaacaGLOa GaayzkaaaaaaqcfaOaayjkaiaawMcaaiaabYgacaqGVbGaae4zaiaa bk7aa8aabaWdbiaaigdacqGHRaWkdaqadaWdaeaapeWaaSaaa8aaba WdbiaabIhaa8aabaWaaCbiaeaapeGaae4SdaWdaeqabaWdbiablkWa KaaaaaaacaGLOaGaayzkaaWdamaaCaaabeqcfasaa8qacaqGJbaaaa aajuaGcqGHsisldaWcaaWdaeaapeGaaeiBaiaab+gacaqGNbGaaeOS diabewGiglaabEcadaqadaWdaeaapeGaaeiEa8aadaWgaaqcfasaa8 qacaqGPbaajuaGpaqabaWdbiaacYcacaqGJbGaaiila8aadaWfGaqa a8qacaqGZoaapaqabeaapeGaeSOadqcaaaGaayjkaiaawMcaaiaabk 7apaWaaWbaaeqajuaibaWdbiabewGigNqbaoaabmaajuaipaqaa8qa caqG4bqcfa4damaaBaaajuaibaWdbiaabMgaa8aabeaapeGaaiilai aabogacaGGSaqcfa4damaaxacajuaibaWdbiaabo7aa8aabeqaa8qa cqWIcmajaaaacaGLOaGaayzkaaaaaaqcfa4daeaapeGaaeOSd8aada ahaaqabKqbGeaapeGaeqybIyCcfa4aaeWaaKqbG8aabaWdbiaabIha juaGpaWaaSbaaKazfa0=baWdbiaabMgaaKqbG8aabeaapeGaaiilai aabogacaGGSaqcfa4damaaxacajuaibaWdbiaabo7aa8aabeqaa8qa cqWIcmajaaaacaGLOaGaayzkaaaaaKqbakabgkHiTiaaigdaaaaaca GLBbGaayzxaaGaaeiOaiaabckacaqGGcGaeyypa0JaaGimaaaa@9D6B@ (5.6)

'( x i ,c, γ ̂ )= ( x i ,c,γ ) γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHfiIXcaqGNaWaaeWaa8aabaWdbiaabIhapaWaaSbaaKqb GeaapeGaaeyAaaqcfa4daeqaa8qacaGGSaGaae4yaiaacYcapaWaaC biaeaapeGaae4SdaWdaeqabaWdbiablkWaKaaaaiaawIcacaGLPaaa cqGH9aqpdaWcaaWdaeaapeGaeyOaIyRaeqybIy8aaeWaa8aabaWdbi aabIhapaWaaSbaaKqbGeaapeGaaeyAaaqcfa4daeqaa8qacaGGSaGa ae4yaiaacYcacaqGZoaacaGLOaGaayzkaaaapaqaa8qacqGHciITca qGZoaaaaaa@511E@

The solution of this system is not possible in a closed form, so by using numerical computation, the solution of the three log-likelihood equations (5.4), (5.5) and (5.6) will provide the MLE of ( β,c,γ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaqadaWdaeaapeGaaeOSdiaacYcacaqGJbGaaiilaiaabo7a aiaawIcacaGLPaaaaaa@3D04@ .

In this study, maximum likelihood estimates of were computed by numerical methods, using the R studio statistical software with the help of “MASS” package. For solving the equations analytically Nelder_Mead optimization method [15] is employed.

We here now consider the four possible cases for estimating the parameters.

Case I: known parameters c and γ and unknown parameter β.

logL β c= c ̂ ,γ= γ ̂ , β= β ̂ =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWcaaWdaeaapeGaeyOaIyRaaeiBaiaab+gacaqGNbGaaeit aaWdaeaapeGaeyOaIyRaaeOSdaaacaqG8bGaaeiOa8aadaWgaaqaa8 qacaqGJbGaeyypa0ZdamaaxacabaWdbiaabogaa8aabeqaa8qacqWI cmajaaGaaiilaiaabo7acqGH9aqppaWaaCbiaeaapeGaae4SdaWdae qabaWdbiablkWaKaaacaGGSaGaaeiOaiaabk7acqGH9aqppaWaaCbi aeaapeGaaeOSdaWdaeqabaWdbiablkWaKaaaa8aabeaacqGH9aqpca aIWaaaaa@52A7@ yields

i=1 n [ log(1+ ( x i /γ) c ) β ^ ϕ( x i ,c,γ) β ^ ϕ( x i ,c,γ)1 ) 1 β ^ ϕ( x i ,c,γ) ) ]=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaabmae aacaGGBbWaaSaaaeaaciGGSbGaai4BaiaacEgacaGGOaGaaGymaiab gUcaRiaacIcacaWG4bWaaSbaaKqbGeaacaWGPbaajuaGbeaacaGGVa Gaeq4SdCMaaiykamaaCaaabeqcfasaaiaadogaaaqcfaOaaiykaaqa aiqbek7aIzaajaaaaiabgkHiTmaalaaabaGaeqy1dyMaaiikaiaadI hadaWgaaqcfasaaiaadMgaaKqbagqaaiaacYcacaWGJbGaaiilaiab eo7aNjaacMcacuaHYoGygaqcamaaCaaabeqcfasaaiabew9aMjaacI cacaWG4bqcfa4aaSbaaKqbGeaacaWGPbaabeaacaGGSaGaam4yaiaa cYcacqaHZoWzcaGGPaGaeyOeI0IaaGymaaaajuaGcaGGPaaabaGaaG ymaiabgkHiTiqbek7aIzaajaWaaWbaaeqajuaibaGaeqy1dyMaaiik aiaadIhajuaGdaWgaaqcfasaaiaadMgaaeqaaiaacYcacaWGJbGaai ilaiabeo7aNjaacMcaaaqcfaOaaiykaaaaaKqbGeaacaWGPbGaeyyp a0JaaGymaaqaaiaad6gaaKqbakabggHiLdGaaiyxaiabg2da9iaaic daaaa@7861@

Solving the Equation (5.7) analytically gives the maximum likelihood estimator β ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqOSdi MbaKaaaaa@3835@ of the parameter β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@ .

Case II: known parameter c and unknown parameters β and γ.

i=1 n [ ( c X i c γ ^ ( c+1 ) )logβ 1+ ( X γ ^ ) c logβ ϕ ' ( X i ,c, γ ^ ) β ϕ( X i ,c, γ ^ ) β ϕ( X i ,c, γ ^ ) 1 ] =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaabCae aadaWadaqaamaalaaabaGaeyOeI0YaaeWaaeaacaWGJbGaamiwamaa DaaajuaibaGaamyAaaqaaiaadogaaaqcfaOafq4SdCMbaKaadaahaa qabKqbGeaacqGHsisljuaGdaqadaqcfasaaiaadogacqGHRaWkcaaI XaaacaGLOaGaayzkaaaaaaqcfaOaayjkaiaawMcaaiGacYgacaGGVb Gaai4zaiabek7aIbqaaiaaigdacqGHRaWkdaqadaqaamaalaaabaGa amiwaaqaaiqbeo7aNzaajaaaaaGaayjkaiaawMcaamaaCaaabeqcfa saaiaadogaaaaaaKqbakabgkHiTmaalaaabaGaciiBaiaac+gacaGG NbGaeqOSdiMaeqy1dy2aaWbaaeqabaGaai4jaaaadaqadaqaaiaadI fadaWgaaqcfasaaiaadMgaaKqbagqaaiaacYcacaWGJbGaaiilaiqb eo7aNzaajaaacaGLOaGaayzkaaGaeqOSdi2aaWbaaeqajuaibaGaeq y1dywcfa4aaeWaaKqbGeaacaWGybqcfa4aaSbaaKqbGeaacaWGPbaa beaacaGGSaGaam4yaiaacYcacuaHZoWzgaqcaaGaayjkaiaawMcaaa aaaKqbagaacqaHYoGydaahaaqabKqbGeaacqaHvpGzjuaGdaqadaqc fasaaiaadIfajuaGdaWgaaqcfasaaiaadMgaaeqaaiaacYcacaWGJb Gaaiilaiqbeo7aNzaajaaacaGLOaGaayzkaaaaaKqbakabgkHiTiaa igdaaaaacaGLBbGaayzxaaaajuaibaGaamyAaiabg2da9iaaigdaae aacaWGUbaajuaGcqGHris5aiabg2da9iaaicdaaaa@884B@ (5.8)

Solving the Equations (5.7) and (5.8) analytically gives the maximum likelihood estimators β Ì‚ and γ Ì‚ of the parameters β and γ.

Case III: known parameter γ and unknown parameters β and c.

logL c c= c ̂ ,γ= γ ̂ , β= β ̂ =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWcaaWdaeaapeGaeyOaIyRaaeiBaiaab+gacaqGNbGaaeit aaWdaeaapeGaeyOaIyRaae4yaaaacaqG8bGaaeiOa8aadaWgaaqaa8 qacaqGJbGaeyypa0ZdamaaxacabaWdbiaabogaa8aabeqaa8qacqWI cmajaaGaaiilaiaabo7acqGH9aqppaWaaCbiaeaapeGaae4SdaWdae qabaWdbiablkWaKaaacaGGSaGaaeiOaiaabk7acqGH9aqppaWaaCbi aeaapeGaaeOSdaWdaeqabaWdbiablkWaKaaaa8aabeaacqGH9aqpca aIWaaaaa@5255@ yields

i=1 n [ (( x i /γ ) c ̂ ) logβlogx i 1+ ( x γ ) c ̂ logβ'( x i , c, ̂ γ ) β ( x i , c ̂ ,γ ) 1 β ( x i , c, ̂ γ ) ]   =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaGfWbqabKqbG8aabaWdbiaabMgacqGH9aqpcaaIXaaapaqa a8qacaqGUbaajuaGpaqaa8qacqGHris5aaWaamWaa8aabaWdbmaala aapaqaa8qacaGGOaWaaeWaa8aabaWdbiaabIhapaWaaSbaaKqbGeaa peGaaeyAaaqcfa4daeqaa8qacaGGVaGaae4SdiaacMcapaWaaWbaae qabaWaaCbiaKqbGeaapeGaae4yaaqcfa4daeqabaWdbiablkWaKaaa aaaacaGLOaGaayzkaaGaaeiBaiaab+gacaqGNbGaaeOSdiaabYgaca qGVbGaae4zaiaabIhapaWaaSbaaKqbGeaapeGaaeyAaaqcfa4daeqa aaqaa8qacaaIXaGaey4kaSYaaeWaa8aabaWdbmaalaaapaqaa8qaca qG4baapaqaa8qacaqGZoaaaaGaayjkaiaawMcaa8aadaahaaqabeaa daWfGaqcfasaa8qacaqGJbaajuaGpaqabeaapeGaeSOadqcaaaaaaa GaeyOeI0YaaSaaa8aabaWdbiaabYgacaqGVbGaae4zaiaabk7acqaH fiIXcaqGNaWaaeWaa8aabaWdbiaabIhapaWaaSbaaeaapeGaaeyAaa Wdaeqaa8qacaGGSaWdamaaxacabaWdbiaabogacaGGSaaapaqabeaa peGaeSOadqcaaiaabo7aaiaawIcacaGLPaaacaqGYoWdamaaCaaabe qcfasaa8qacqaHfiIXjuaGdaqadaqcfaYdaeaapeGaaeiEaKqba+aa daWgaaqcfasaa8qacaqGPbaapaqabaWdbiaacYcajuaGpaWaaCbiaK qbGeaapeGaae4yaaWdaeqabaWdbiablkWaKaaacaGGSaGaae4SdaGa ayjkaiaawMcaaaaaaKqba+aabaWdbiaaigdacqGHsislcaqGYoWdam aaCaaabeqcfasaa8qacqaHfiIXjuaGdaqadaqcfaYdaeaapeGaaeiE aKqba+aadaWgaaqcfasaa8qacaqGPbaapaqabaWdbiaacYcajuaGpa WaaCbiaKqbGeaapeGaae4yaiaacYcaa8aabeqaa8qacqWIcmajaaGa ae4SdaGaayjkaiaawMcaaaaaaaaajuaGcaGLBbGaayzxaaGaaeiOai aabckacaqGGcGaeyypa0JaaGimaaaa@9121@ (5.9)

Solving the Equations (5.7) and (5.9) analytically gives the maximum likelihood estimators β Ì‚ and c Ì‚ of the parameters and .

Case IV: Unknown parameters β , c and γ . Solving the Equations (5.7), (5.8) and (5.9) analytically gives the maximum likelihood estimators c ̂  ,  γ ̂   and  β ̂ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaCbiae aaqaaaaaaaaaWdbiaabogaa8aabeqaa8qacqWIcmajaaGaaeiOaiaa cYcapaWaaCbiaeaapeGaaeiOaiaabo7aa8aabeqaa8qacqWIcmajaa GaaeiOaiaabckacaqGHbGaaeOBaiaabsgacaqGGcWdamaaxacabaWd biaabk7aa8aabeqaa8qacqWIcmajaaaaaa@4699@ of the parameters c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGJbaaaa@378D@ , γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHZoWzaaa@384C@ and β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@ respectively.

Estimation of the parameters based on the proportion method: Khan et al. [16] proposed and provided a motivation for the method of proportions to estimate the parameters for discrete Weibull distribution. Now, we present a similar method for the three parameter discrete Burr type XII distribution and discrete Lomax distribution for the same reasons as outlined [16]. Let x 1, x 2 , x 3, , x n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqG4bWdamaaBaaajuaibaWdbiaaigdacaGGSaaajuaGpaqa baWdbiaabIhapaWaaSbaaKqbGeaapeGaaGOmaaqcfa4daeqaa8qaca GGSaGaaeiEa8aadaWgaaqcfasaa8qacaaIZaGaaiilaaqcfa4daeqa a8qacqGHMacVcaGGSaGaaeiEa8aadaWgaaqcfasaa8qacaqGUbaaju aGpaqabaaaaa@4660@ be a random sample from the distribution with pmf (3.1). Define the indicator function by I u ( x i )={ 1            if  x i =u  0            if  x i u  MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGjbWdamaaBaaajuaibaWdbiaadwhaaKqba+aabeaapeWa aeWaa8aabaWdbiaabIhapaWaaSbaaKqbGeaapeGaaeyAaaqcfa4dae qaaaWdbiaawIcacaGLPaaacqGH9aqpdaGabaWdaeaafaqabeGabaaa baWdbiaaigdacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckaca GGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaWGPbGaamOzaiaa cckacaqG4bWdamaaBaaajuaibaWdbiaabMgaaKqba+aabeaapeGaey ypa0JaamyDaiaacckaa8aabaWdbiaaicdacaGGGcGaaiiOaiaaccka caGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOai aacckacaWGPbGaamOzaiaacckacaqG4bWdamaaBaaajuaibaWdbiaa bMgaaKqba+aabeaapeGaeyiyIKRaamyDaiaacckaaaaacaGL7baaaa a@7079@

Denote f u = I u ( x i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGMbWdamaaBaaajuaibaWdbiaabwhaaKqba+aabeaapeGa eyypa0ZdamaavacabeqabeaacaaMb8oabaWdbiabggHiLdaacaWGjb WdamaaBaaajuaibaWdbiaadwhaaKqba+aabeaapeWaaeWaa8aabaWd biaabIhapaWaaSbaaKqbGeaapeGaaeyAaaqcfa4daeqaaaWdbiaawI cacaGLPaaaaaa@45C1@ by the frequency of the value u in the observed sample.

Therefore, the proportion (relative frequency) R u = f u n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGsbWdamaaBaaajuaibaWdbiaadwhaaKqba+aabeaapeGa eyypa0ZaaSaaa8aabaWdbiaadAgapaWaaSbaaKqbGeaapeGaamyDaa qcfa4daeqaaaqaa8qacaWGUbaaaaaa@3EB9@ can be used to estimate the probability P( u;β,c,γ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGqbWaaeWaa8aabaWdbiaadwhacaGG7aGaeqOSdiMaaiil aiaadogacaGGSaGaae4SdaGaayjkaiaawMcaaaaa@3FFD@ . Now we consider the following cases for the purpose of parameter estimation.

Case I: known parameters c and γ and unknown parameter β.

This is the simplest case. The unknown parameter β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@ has a proportion estimator in exact solution, where

P( 0;β,c,γ )= 1 β log( 1+ 1 γ c )  = f 0 n      Where 0<β<1;γ>0;c>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGqbWaaeWaa8aabaWdbiaaicdacaGG7aGaeqOSdiMaaiil aiaadogacaGGSaGaae4SdaGaayjkaiaawMcaaiabg2da9iaabckaca aIXaGaeyOeI0IaaeOSd8aadaahaaqabKqbGeaapeGaciiBaiaac+ga caGGNbqcfa4aaeWaaKqbG8aabaWdbiaaigdacqGHRaWkjuaGdaWcaa qcfaYdaeaapeGaaGymaaWdaeaapeGaae4SdKqba+aadaahaaqcfasa beaapeGaae4yaaaaaaaacaGLOaGaayzkaaaaaKqbakaabckacqGH9a qpdaWcaaWdaeaapeGaamOza8aadaWgaaqcfasaa8qacaaIWaaajuaG paqabaaabaWdbiaad6gaaaGaaeiOaiaabckacaqGGcGaaeiOaiaabc kacaqGxbGaaeiAaiaabwgacaqGYbGaaeyzaiaabckacaaIWaGaeyip aWJaeqOSdiMaeyipaWJaaGymaiaacUdacqaHZoWzcqGH+aGpcaaIWa Gaai4oaiaadogacqGH+aGpcaaIWaaaaa@6F8B@ (5.10)

β * = e log( ( 1 f 0 n ) ( 1 1 γ c ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHYoGypaWaaWbaaeqabaWdbiaacQcaaaGaeyypa0Jaamyz a8aadaahaaqabKqbGeaapeGaamiBaiaad+gacaWGNbqcfa4aaeWaaK qbG8aabaqcfa4dbmaalaaajuaipaqaaKqba+qadaqadaqcfaYdaeaa peGaaGymaiabgkHiTKqbaoaalaaajuaipaqaa8qacaWGMbqcfa4dam aaBaaajuaibaWdbiaaicdaa8aabeaaaeaapeGaamOBaaaaaiaawIca caGLPaaaa8aabaqcfa4dbmaabmaajuaipaqaa8qacaaIXaGaeyOeI0 scfa4aaSaaaKqbG8aabaWdbiaaigdaa8aabaWdbiaabo7ajuaGpaWa aWbaaKqbGeqabaWdbiaadogaaaaaaaGaayjkaiaawMcaaaaaaiaawI cacaGLPaaaaaaaaa@536C@

f 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGMbWdamaaBaaajuaibaWdbiaaicdaaKqba+aabeaaaaa@3955@ denotes the number of zero’s in a sample of size n.

Case II: known parameter c and unknown parameters β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@ and γ.

Let f 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGMbWdamaaBaaajuaibaWdbiaaigdaaKqba+aabeaaaaa@3956@ denote the number of one’s in the sample of size n.

β log( 1+ 1 γ c ) β log( 1+ ( 2 γ ) c ) = f 1 n    Where 0<β<1;γ>0;c>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGYoWdamaaCaaabeqcfasaa8qaciGGSbGaai4BaiaacEga juaGdaqadaqcfaYdaeaapeGaaGymaiabgUcaRKqbaoaalaaajuaipa qaa8qacaaIXaaapaqaa8qacaqGZoqcfa4damaaCaaajuaibeqaa8qa caqGJbaaaaaaaiaawIcacaGLPaaaaaqcfaOaeyOeI0IaaeOSd8aada ahaaqabKqbGeaapeGaciiBaiaac+gacaGGNbqcfa4aaeWaaKqbG8aa baWdbiaaigdacqGHRaWkjuaGdaqadaqcfaYdaeaajuaGpeWaaSaaaK qbG8aabaWdbiaaikdaa8aabaWdbiaabo7aaaaacaGLOaGaayzkaaqc fa4damaaCaaajuaibeqaa8qacaqGJbaaaaGaayjkaiaawMcaaaaaju aGcqGH9aqpdaWcaaWdaeaapeGaamOza8aadaWgaaqcfasaa8qacaaI XaaajuaGpaqabaaabaWdbiaad6gaaaGaaeiOaiaabckacaqGGcGaae 4vaiaabIgacaqGLbGaaeOCaiaabwgacaqGGcGaaGimaiabgYda8iab ek7aIjabgYda8iaaigdacaGG7aGaeq4SdCMaeyOpa4JaaGimaiaacU dacaWGJbGaeyOpa4JaaGimaaaa@7073@ (5.11)

Solving equations (5.10) and (5.11) numerically using Newton_Raphson (N_R) method, gives the proportion estimators β * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHYoGypaWaaWbaaeqabaWdbiaacQcaaaaaaa@3935@ and γ * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGZoWdamaaCaaabeqaa8qacaqGQaaaaaaa@38CC@ of the parameters β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3826@ and γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC gaaa@382C@ .

Case III: known parameter γ and unknown parameters β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3826@ and c. Solving equations (5.10) and (5.11) numerically using Newton_Raphson (N_R) method, gives the proportion estimators β* MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHYoGycaGGQaaaaa@38F4@ and c* MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4yai aacQcaaaa@381A@ of the parameters β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3826@ and γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGZoaaaa@37DE@ .

Let f 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGMbWdamaaBaaajuaibaWdbiaaikdaaKqba+aabeaaaaa@3957@ denote the number of two’s in the sample of size n.

β log( 1+ ( 2 γ ) c ) β log( 1+ ( 3 γ ) c ) = f 2 n    Where 0<β<1;γ>0;c>0  MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGYoWdamaaCaaabeqcfasaa8qaciGGSbGaai4BaiaacEga juaGdaqadaqcfaYdaeaapeGaaGymaiabgUcaRKqbaoaabmaajuaipa qaaKqba+qadaWcaaqcfaYdaeaapeGaaGOmaaWdaeaapeGaae4Sdaaa aiaawIcacaGLPaaajuaGpaWaaWbaaKqbGeqabaWdbiaabogaaaaaca GLOaGaayzkaaaaaKqbakabgkHiTiaabk7apaWaaWbaaeqajuaibaWd biGacYgacaGGVbGaai4zaKqbaoaabmaajuaipaqaa8qacaaIXaGaey 4kaSscfa4aaeWaaKqbG8aabaqcfa4dbmaalaaajuaipaqaa8qacaaI Zaaapaqaa8qacaqGZoaaaaGaayjkaiaawMcaaKqba+aadaahaaqcfa sabeaapeGaae4yaaaaaiaawIcacaGLPaaaaaqcfaOaeyypa0ZaaSaa a8aabaWdbiaadAgapaWaaSbaaKqbGeaapeGaaGOmaaqcfa4daeqaaa qaa8qacaWGUbaaaiaabckacaqGGcGaaeiOaiaabEfacaqGObGaaeyz aiaabkhacaqGLbGaaeiOaiaaicdacqGH8aapcqaHYoGycqGH8aapca aIXaGaai4oaiabeo7aNjabg6da+iaaicdacaGG7aGaam4yaiabg6da +iaaicdacaGGGcaaaa@73FE@ (5.12)

Solving the Equations (5.10), (5.11) and (5.12) analytically using Newton_Raphson (N_R) method, gives the proportion estimators β * ,  c * and  γ *   MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHYoGypaWaaWbaaeqabaWdbiaacQcaaaGaaiilaiaabcka caqGJbWdamaaCaaabeqaa8qacaqGQaaaaiaabggacaqGUbGaaeizai aabckacaqGGcGaae4Sd8aadaahaaqabeaapeGaaeOkaaaacaqGGcaa aa@4528@ of the parameters β,c and γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi MaaiilaiaadogaqaaaaaaaaaWdbiaacckapaGaamyyaiaad6gacaWG KbWdbiaacckapaGaeq4SdCgaaa@40BC@ respectively.

Some Theorems Related to Three Parameter Discrete Burr Type XII Distribution and Discrete Lomax Distribution

In this section we discuss some important theorems which relate three parameter discrete Burr type XII Distribution and discrete Lomax distribution with some important discrete class of continuous distributions already in the literature.

Theorem 1: Let X be random variable following three parameter continuous Burr XII distribution with E( X r )<      r=1,2,3. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGfbWaaeWaa8aabaWdbiaabIfapaWaaWbaaeqajuaibaWd biaabkhaaaaajuaGcaGLOaGaayzkaaGaeyipaWJaeyOhIuQaaiiOai aacckacaGGGcGaaiiOaiaacckacqGHaiIicaGGGcGaamOCaiabg2da 9iaaigdacaGGSaGaaGOmaiaacYcacaaIZaGaeyOjGWRaaiOlaaaa@4DD0@ Then E( Y r )<  MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGfbWaaeWaa8aabaWdbiaabMfapaWaaWbaaeqajuaibaWd biaabkhaaaaajuaGcaGLOaGaayzkaaGaeyipaWJaaiiOaiabg6HiLc aa@3F7C@ where Y=[ X ]~DBX( x;β,c,γ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGzbGaeyypa0ZaamWaa8aabaWdbiaabIfaaiaawUfacaGL DbaacaGG+bGaaeiraiaabkeacaqGybWaaeWaa8aabaWdbiaadIhaca GG7aGaeqOSdiMaaiilaiaadogacaGGSaGaae4SdaGaayjkaiaawMca aaaa@4762@

Proof: Proof is straight forward, since 0[ X ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaaIWaGaeyizIm6aamWaa8aabaWdbiaabIfaaiaawUfacaGL DbaacqGHKjYOcaqGybGaaeiOaaaa@3FB3@ , so clearly if E( X r )<           r=1,2,3....... MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGfbGaaiikaiaadIfapaWaaWbaaeqajuaibaWdbiaadkha aaqcfaOaaiykaiabgYda8iabg6HiLkaacckacaGGGcGaaiiOaiaacc kacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaeyia IiIaamOCaiabg2da9iaaigdacaGGSaGaaGOmaiaacYcacaaIZaGaai Olaiaac6cacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaaaa@55D9@ Then E( [X] r )< MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGfbWaaeWaa8aabaWdbiaacUfacaqGybGaaiyxa8aadaah aaqabKqbGeaapeGaaeOCaaaaaKqbakaawIcacaGLPaaacqGH8aapcq GHEisPaaa@4017@ .

Theorem 2: If X~DBDXII( x;β,c,γ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGybGaaiOFaiaadseacaWGcbGaamiraiabgkHiTiaadIfa caWGjbGaamysamaabmaabaGaamiEaiaacUdacqaHYoGycaGGSaGaam 4yaiaacYcacqaHZoWzaiaawIcacaGLPaaaaaa@4717@ then Y=[ [ log( 1+ ( x γ ) c ) ] 1/c ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGzbGaeyypa0ZaamWaa8aabaWdbmaadmaapaqaa8qacaqG SbGaae4BaiaabEgadaqadaWdaeaapeGaaGymaiabgUcaRmaabmaapa qaa8qadaWcaaWdaeaapeGaaeiEaaWdaeaapeGaae4SdaaaaiaawIca caGLPaaapaWaaWbaaeqajuaibaWdbiaabogaaaaajuaGcaGLOaGaay zkaaaacaGLBbGaayzxaaWdamaaCaaabeqcfasaa8qacqGHsislcaaI XaGaai4laiaabogaaaaajuaGcaGLBbGaayzxaaaaaa@4D04@ follows discrete inverse Weibull distribution i.e., DIW ( c,β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGJbGaaiilaiabek7aIbaa@39DE@ )

β= e k  ;  0<β<1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGYoGaeyypa0Jaaeyza8aadaahaaqabKqbGeaapeGaeyOe I0Iaae4AaaaajuaGcaqGGcGaai4oaiaabckacaqGGcGaaGimaiabgY da8iabek7aIjabgYda8iaaigdaaaa@45E9@

Proof:-

P[ Yy ]=P[ [ [ log( 1+ ( x γ ) c ) ] 1/c ]y ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGqbWaamWaa8aabaWdbiaabMfacqGHLjYScaqG5baacaGL BbGaayzxaaGaeyypa0Jaaeiuamaadmaapaqaa8qadaWadaWdaeaape WaamWaa8aabaWdbiaabYgacaqGVbGaae4zamaabmaapaqaa8qacaaI XaGaey4kaSYaaeWaa8aabaWdbmaalaaapaqaa8qacaqG4baapaqaa8 qacaqGZoaaaaGaayjkaiaawMcaa8aadaahaaqabKqbGeaapeGaae4y aaaaaKqbakaawIcacaGLPaaaaiaawUfacaGLDbaapaWaaWbaaeqaju aibaWdbiabgkHiTiaaigdacaGGVaGaae4yaaaaaKqbakaawUfacaGL DbaacqGHLjYScaqG5baacaGLBbGaayzxaaaaaa@5850@

=P[ [ log( 1+ ( x γ ) c ) ] 1/c y ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGH9aqpcaqGqbWaamWaa8aabaWdbmaadmaapaqaa8qacaqG SbGaae4BaiaabEgadaqadaWdaeaapeGaaGymaiabgUcaRmaabmaapa qaa8qadaWcaaWdaeaapeGaaeiEaaWdaeaapeGaae4SdaaaaiaawIca caGLPaaapaWaaWbaaeqajuaibaWdbiaabogaaaaajuaGcaGLOaGaay zkaaaacaGLBbGaayzxaaWdamaaCaaabeqcfasaa8qacqGHsislcaaI XaGaai4laiaabogaaaqcfaOaeyyzImRaaeyEaaGaay5waiaaw2faaa aa@4FBD@

=P[ X [ γ c ( e y c 1)] 1/c ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGH9aqpcaqGqbWaamWaa8aabaWdbiaabIfacqGHLjYScaGG BbGaae4Sd8aadaahaaqabKqbGeaapeGaae4yaaaajuaGcaGGOaGaae yza8aadaahaaqabKqbGeaapeGaaeyEaKqba+aadaahaaqcfasabeaa peGaeyOeI0Iaae4yaaaaaaqcfaOaeyOeI0IaaGymaiaacMcacaGGDb WdamaaCaaabeqcfasaa8qacaaIXaGaai4laiaabogaaaaajuaGcaGL BbGaayzxaaaaaa@4E0F@

=1 β log[ 1+ [ [ γ c ( e y c 1)] 1/c γ ] c ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGH9aqpcaaIXaGaeyOeI0IaaeOSd8aadaahaaqabKqbGeaa peGaaeiBaiaab+gacaqGNbqcfa4aamWaaKqbG8aabaWdbiaaigdacq GHRaWkjuaGdaWadaqcfaYdaeaajuaGpeWaaSaaaKqbG8aabaWdbiaa cUfacaqGZoqcfa4damaaCaaajuaibeqaa8qacaqGJbaaaiaacIcaca qGLbqcfa4damaaCaaajuaibeqaa8qacaqG5bqcfa4damaaCaaajuai beqaa8qacqGHsislcaqGJbaaaaaacqGHsislcaaIXaGaaiykaiaac2 fajuaGpaWaaWbaaKqbGeqabaWdbiaaigdacaGGVaGaae4yaaaaa8aa baWdbiaabo7aaaaacaGLBbGaayzxaaqcfa4damaaCaaajuaibeqaa8 qacaqGJbaaaaGaay5waiaaw2faaaaaaaa@5A01@

=1 β log e y c =1 β y c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGH9aqpcaaIXaGaeyOeI0IaaeOSd8aadaahaaqabKqbGeaa peGaciiBaiaac+gacaGGNbGaaeyzaKqba+aadaahaaqcfasabeaape GaaeyEaKqba+aadaahaaqcfasabeaapeGaeyOeI0Iaae4yaaaaaaaa aKqbakabg2da9iaaigdacqGHsislcaqGYoWdamaaCaaabeqcfasaa8 qacaqG5bqcfa4damaaCaaajuaibeqaa8qacqGHsislcaqGJbaaaaaa aaa@4C2A@

Which is the survival function of a discrete inverse Weibull distribution.

Hence Y~ DIW( c, β ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaGGzbGaaiOFaiaacckacaWGebGaamysaiaadEfadaqadaWd aeaapeGaam4yaiaacYcacaGGGcGaeqOSdigacaGLOaGaayzkaaaaaa@4220@

Theorem3: If X DBDXII (x;β,c,γ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGybGaaiiOaiablYJi6iaadseacaWGcbGaamiraiabgkHi TiaadIfacaWGjbGaamysaiaabccapaGaaiika8qacaqG4bGaai4oai aabk7acaGGSaGaae4yaiaacYcacaqGZoWdaiaacMcaaaa@4828@ then Y=[ log[ 1+ ( x γ ) c ] ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGzbGaeyypa0ZaamWaa8aabaWdbmaakaaapaqaa8qacaqG SbGaae4BaiaabEgadaWadaWdaeaapeGaaGymaiabgUcaRmaabmaapa qaa8qadaWcaaWdaeaapeGaaeiEaaWdaeaapeGaae4SdaaaaiaawIca caGLPaaapaWaaWbaaeqajuaibaWdbiaabogaaaaajuaGcaGLBbGaay zxaaaabeaaaiaawUfacaGLDbaaaaa@474D@ follows discrete Raleigh distribution i.e., DRel ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGOaGaaeOSdiaacMcaaaa@3935@

β= e k  ;  0<β<1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGYoGaeyypa0Jaaeyza8aadaahaaqabKqbGeaapeGaeyOe I0Iaae4AaaaajuaGcaqGGcGaai4oaiaabckacaqGGcGaaGimaiabgY da8iabek7aIjabgYda8iaaigdaaaa@45E9@

Proof:-

P[ Yy ]=P[ [ log[ 1+ ( x γ ) c ] ]y ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGqbWaamWaa8aabaWdbiaabMfacqGHLjYScaqG5baacaGL BbGaayzxaaGaeyypa0Jaaeiuamaadmaapaqaa8qadaWadaWdaeaape WaaOaaa8aabaWdbiaabYgacaqGVbGaae4zamaadmaapaqaa8qacaaI XaGaey4kaSYaaeWaa8aabaWdbmaalaaapaqaa8qacaqG4baapaqaa8 qacaqGZoaaaaGaayjkaiaawMcaa8aadaahaaqabKqbGeaapeGaae4y aaaaaKqbakaawUfacaGLDbaaaeqaaaGaay5waiaaw2faaiabgwMiZk aabMhaaiaawUfacaGLDbaaaaa@5299@

=P[ X[ γ c ( e y 2 1 ) ] 1/c ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGH9aqpcaqGqbWaamWaa8aabaWdbiaabIfacqGHLjYSdaqc saWdaeaapeGaae4Sd8aadaahaaqabKqbGeaapeGaae4yaaaajuaGca GGOaGaaeyza8aadaahaaqabKqbGeaapeGaaeyEaKqba+aadaahaaqc fasabeaapeGaaGOmaaaaaaqcfaOaeyOeI0IaaGymaaGaay5waiaawM caaiaac2fapaWaaWbaaeqajuaibaWdbiaaigdacaGGVaGaae4yaaaa aKqbakaawUfacaGLDbaaaaa@4D5E@

= β log[ 1+ [ [ γ c ( e y 2 1 ) ] 1/c γ ] c ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGH9aqpcqaHYoGydaahaaqabeaaciGGSbGaai4BaiaacEga daWadaqaaiaaigdacqGHRaWkdaWadaqaamaalaaabaWaamWaaeaacq aHZoWzdaahaaqabKqbGeaacaWGJbaaaKqbaoaabmaabaGaamyzamaa CaaabeqcfasaaiaadMhajuaGdaahaaqcfasabeaacaaIYaaaaaaaju aGcqGHsislcaaIXaaacaGLOaGaayzkaaaacaGLBbGaayzxaaWaaWba aeqajuaibaGaaGymaiaac+cacaWGJbaaaaqcfayaaiabeo7aNbaaai aawUfacaGLDbaadaahaaqabKqbGeaacaWGJbaaaaqcfaOaay5waiaa w2faaaaaaaa@555F@

= β log e y 2 = β y 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGH9aqpcaqGYoWdamaaCaaabeqcfasaa8qaciGGSbGaai4B aiaacEgacaqGLbqcfa4damaaCaaajuaibeqaa8qacaqG5bqcfa4dam aaCaaajuaibeqaa8qacaaIYaaaaaaaaaqcfaOaeyypa0JaaeOSd8aa daahaaqabKqbGeaapeGaaeyEaKqba+aadaahaaqcfasabeaapeGaaG Omaaaaaaaaaa@46AC@

which is the survival function of a discrete Raleigh distribution. Hence Y~ DRaleigh(  β ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaGGzbGaaiOFaiaacckacaWGebGaamOuaiaadggacaWGSbGa amyzaiaadMgacaWGNbGaamiAamaabmaapaqaa8qacaGGGcGaeqOSdi gacaGLOaGaayzkaaaaaa@453D@ . Corollary. If X DLomax (x;β,γ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGybGaeSipIOJaaiiOaiaadseacaWGmbGaam4Baiaad2ga caWGHbGaamiEaiaabccapaGaaiikaiaacIhacaGG7aGaeqOSdiMaai ilaiabeo7aNjaacMcaaaa@46EF@ then Y=[ log[ 1+( x γ ) ] ]~DRaleigh( β ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGzbGaeyypa0ZaamWaa8aabaWdbmaakaaapaqaa8qacaqG SbGaae4BaiaabEgadaWadaWdaeaapeGaaGymaiabgUcaRmaabmaapa qaa8qadaWcaaWdaeaapeGaaeiEaaWdaeaapeGaae4SdaaaaiaawIca caGLPaaaaiaawUfacaGLDbaaaeqaaaGaay5waiaaw2faaiaac6haca WGebGaamOuaiaadggacaWGSbGaamyzaiaadMgacaWGNbGaamiAamaa bmaapaqaa8qacqaHYoGyaiaawIcacaGLPaaaaaa@50DD@

Application of Discrete Lomax Distribution and Three Parameter Discrete Burr Type XII Distribution in Medical Science

Here we consider the data set of counts of cysts of kidneys using steroids as given in the Table 3. The example data set originated from a study [1] investigating the effect of a corticosteroid on cyst formation in mice fetuses undertaken within the Department of Nephro-Urology at the Institute of Child Health of University College London. Embryonic mouse kidneys were cultured, and a random sample was subjected to steroids (110). Table 4 exhibits some descriptive measures of count data of cysts of kidneys using steroids based on 1000 bootstrap samples.

For the purpose of parameter estimation, we employ the fitdistr procedure in R studio statistical software to find out the estimates of the parameters. The ML estimates and their standard errors provided by the fitdistr procedure are given in the Table 5. In Figure 20 the empirical cdf of the number of cysts in a kidney using steroid has been shown.

We compute the expected frequencies for fitting discrete Lomax, three parameter discrete Burr type XII, Poisson, Geometric, Inflated Poisson and DRayleigh distributions with the help of R studio statistical software and Pearson’s chi-square test is applied to check the goodness of fit of the models discussed. The calculated figures are given in the Table 5.

The p-values of Pearson’s Chi-square statistic are 0.532, 0.352, 0.0008, 0.000, 0.000 and 0.0006 for three parameter discrete Burr type XII, discrete Lomax, Zero Inflated Poisson, Poisson, Discrete Raleigh and geometric distributions, respectively Table 6. This reveals that Zero Inflated Poisson, Poisson, Geometric and discrete Rayleigh distributions are not good fit at all, whereas three parameter Burr type XII distribution and two parameter discrete Lomax distributions are good fit distributions with three parameter discrete Burr type XII model being the best one. The null hypothesis that data come from three parameter Burr type XII and two parameter discrete Lomax distributions is accepted. Figure 21 exhibits the graphical overview of the fitted distributions.

We have compared three parameter discrete Burr type XII distribution and two parameter discrete Lomax distribution with discrete Raleigh, Poisson, Zero Inflated Poisson and Geometric distributions using the Akaike information criterion (AIC), given by Akaike [17] and the Bayesian information criterion (BIC), given by Schwarz [18]. Generic function calculating Akaike's ‘An Information Criterion’ for one or several fitted model objects for which a log-likelihood value can be obtained, according to the formula -2*log-likelihood + k*npar, where npar represents the number of parameters in the fitted model, and k = 2 for the usual AIC, or k = log(n) (n being the number of observations) for the so called BIC or SBC (Schwarz's Bayesian criterion).

From Table 7, it is obvious that AIC and BIC criterion favors three parameter discrete Burr type XII and two parameter Lomax distributions in comparison with the Poisson, Zero Inflated Poisson, discrete Raleigh and Geometric distributions, in the case of Counts of cysts of kidneys using steroids.

Figure 21 exhibits graphical overview of the AIC, BIC and negative loglikelihood values for fitted distributions.

Counts of Cysts of Kidneys using Steroids

0

1

2

3

4

5

6

7

8

9

10

11

Total

Frequency

65

14

10

6

4

2

2

2

1

1

1

2

110

Table 3: Counts of cysts of kidneys using steroids.           

Descriptive Measures

Statistic

Standard Error

Bootstrapa

Bias

Standard Error

95% Confidence Interval

Lower

Upper

Sum

153

Mean

1.39

0.236

0.01

0.23

0.95

1.87

Standard Deviation

2.472

-0.018

0.309

1.812

3.053

Variance

6.112

0.009

1.511

3.283

9.324

Skewness

2.293

0.23

-0.04

0.308

1.685

2.908

Kurtosis

5.089

0.457

-0.069

1.911

1.963

9.531

Valid N (listwise)

N

110

0

0

110

110

a. Bootstrap results are based on 1000 bootstrap samples

Table 4: Descriptive statistics of Counts of cysts of kidneys using steroids. 

Distribution

Parameter Estimates

Standard Error of the Estimates

Model Function

Discrete Lomax

β=0.15,γ=1.83 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi Maeyypa0JaaGimaiaac6cacaaIXaGaaGynaiaacYcacqaHZoWzcqGH 9aqpcaaIXaGaaiOlaiaaiIdacaaIZaaaaa@425A@

[0.098, 0.953]

β log( 1+( x y ) ) β log( 1+( x+1 γ ) )       x=0,1,2,3.... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaWbaaeqajuaibaGaciiBaiaac+gacaGGNbqcfa4aaeWaaKqbGeaa caaIXaGaey4kaSscfa4aaeWaaKqbGeaajuaGdaWcaaqcfasaaiaadI haaeaacaWG5baaaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaajuaG cqGHsislcqaHYoGydaahaaqabKqbGeaaciGGSbGaai4BaiaacEgaju aGdaqadaqcfasaaiaaigdacqGHRaWkjuaGdaqadaqcfasaaKqbaoaa laaajuaibaGaamiEaiabgUcaRiaaigdaaeaacqaHZoWzaaaacaGLOa GaayzkaaaacaGLOaGaayzkaaaaaKqbacbaaaaaaaaapeGaaiiOaiaa cckacaGGGcGaaiiOaiaacckacaGGGcGaaiiEaiabg2da9iaaicdaca GGSaGaaGymaiaacYcacaaIYaGaaiilaiaaiodacaGGUaGaaiOlaiaa c6cacaGGUaaaaa@66F1@ where β= e k ;0<β<1;γ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi Maeyypa0JaamyzamaaCaaabeqcfasaaiabgkHiTiaadUgaaaqcfaOa ai4oaiaaicdacqGH8aapcqaHYoGycqGH8aapcaaIXaGaai4oaiabeo 7aNjabg6da+iaaicdaaaa@46D5@

Poisson

λ=1.39 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4UdW Maeyypa0JaaGymaiaac6cacaaIZaGaaGyoaaaa@3C2B@

[0.112]

e λ λ x x! λ>0;x=0,1,2,........ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGLbWaaWbaaeqajuaibaGaeyOeI0Iaeq4UdWgaaKqbakabeU7a SnaaCaaabeqcfasaaiaadIhaaaaajuaGbaGaamiEaiaacgcaaaGaeq 4UdWMaeyOpa4JaaGimaiaacUdacaWG4bGaeyypa0JaaGimaiaacYca caaIXaGaaiilaiaaikdacaGGSaGaaiOlaiaac6cacaGGUaGaaiOlai aac6cacaGGUaGaaiOlaiaac6caaaa@5037@

DRayleigh

q=0.90

[0.009]

q x 2 q ( x+1 ) 2    0<q<1;x=0,1,2,..... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyCam aaCaaabeqcfasaaiaadIhajuaGdaahaaqcfasabeaacaaIYaaaaaaa juaGcqGHsislcaWGXbWaaWbaaKqbGeqabaqcfa4aaeWaaKqbGeaaca WG4bGaey4kaSIaaGymaaGaayjkaiaawMcaaKqbaoaaCaaajuaibeqa aiaaikdaaaaaaKqbacbaaaaaaaaapeGaaiiOaiaacckacaGGGcGaaG imaiabgYda8iaadghacqGH8aapcaaIXaGaai4oaiaadIhacqGH9aqp caaIWaGaaiilaiaaigdacaGGSaGaaGOmaiaacYcacaGGUaGaaiOlai aac6cacaGGUaGaaiOlaaaa@56A5@

Geom

q=0.418

[0.03]

q x q ( x+1 )    0<q<1;x=0,1,2,..... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGXbWaaWbaaeqajuaibaGaamiEaaaajuaGcqGHsislcaWG XbWaaWbaaKqbGeqabaqcfa4aaeWaaKqbGeaacaWG4bGaey4kaSIaaG ymaaGaayjkaiaawMcaaaaajuaGcaGGGcGaaiiOaiaacckacaaIWaGa eyipaWJaamyCaiabgYda8iaaigdacaGG7aGaamiEaiabg2da9iaaic dacaGGSaGaaGymaiaacYcacaaIYaGaaiilaiaac6cacaGGUaGaaiOl aiaac6cacaGGUaaaaa@5371@

Three Parameter Burr type XII

β=0.003,c=0.72,γ=12.75 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi Maeyypa0JaaGimaiaac6cacaaIWaGaaGimaiaaiodacaGGSaGaam4y aiabg2da9iaaicdacaGGUaGaaG4naiaaikdacaGGSaGaeq4SdCMaey ypa0JaaGymaiaaikdacaGGUaGaaG4naiaaiwdaaaa@4955@

[0.002, 0.087, 5.06]

β log( 1+ ( x/γ ) c ) β log( 1+( x+1 )/γ ) c )    x=0,1,2,... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHYoGydaahaaqabKqbGeaaciGGSbGaai4BaiaacEgajuaG daqadaqcfasaaiaaigdacqGHRaWkjuaGdaqadaqcfasaaiaadIhaca GGVaGaeq4SdCgacaGLOaGaayzkaaqcfa4aaWbaaKqbGeqabaGaam4y aaaaaiaawIcacaGLPaaaaaqcfaOaeyOeI0IaeqOSdi2aaWbaaeqaju aibaGaciiBaiaac+gacaGGNbqcfa4aaeWaaKqbGeaacaaIXaGaey4k aSscfa4aaeWaaKqbGeaacaWG4bGaey4kaSIaaGymaaGaayjkaiaawM caaiaac+cacqaHZoWzcaGGPaqcfa4aaWbaaKqbGeqabaGaam4yaaaa aiaawIcacaGLPaaaaaqcfaOaaiiOaiaacckacaGGGcGaamiEaiabg2 da9iaaicdacaGGSaGaaGymaiaacYcacaaIYaGaaiilaiaac6cacaGG UaGaaiOlaaaa@66C7@ where 0<β<1;c>0;γ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaaIWaGaeyipaWJaeqOSdiMaeyipaWJaaGymaiaacUdacaWG JbGaeyOpa4JaaGimaiaacUdacqaHZoWzcqGH+aGpcaaIWaaaaa@4353@   

Zero Inflated Poisson

α=3.27,0.57 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde Maeyypa0JaaG4maiaac6cacaaIYaGaaG4naiaacYcacaaIWaGaaiOl aiaaiwdacaaI3aaaaa@3FB1@

[0.049, 0.283]

{ α+( 1α ) e λ λ x /x!λ>0;x=0 ( 1α ) e λ λ x /x!λ>0;x=1,2,..... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiqaaq aabeqaaiabeg7aHjabgUcaRmaabmaabaGaaGymaiabgkHiTiabeg7a HbGaayjkaiaawMcaaiaadwgadaahaaqabKqbGeaacqGHsislcqaH7o aBaaqcfaOaeq4UdW2aaWbaaeqajuaibaGaamiEaaaajuaGcaGGVaGa amiEaiaacgcacqaH7oaBcqGH+aGpcaaIWaGaai4oaiaadIhacqGH9a qpcaaIWaaabaWaaeWaaeaacaaIXaGaeyOeI0IaeqySdegacaGLOaGa ayzkaaGaamyzamaaCaaabeqcfasaaiabgkHiTiabeU7aSbaajuaGcq aH7oaBdaahaaqabKqbGeaacaWG4baaaKqbakaac+cacaWG4bGaaiyi aiabeU7aSjabg6da+iaaicdacaGG7aGaamiEaiabg2da9iaaigdaca GGSaGaaGOmaiaacYcacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaaaa caGL7baaaaa@6BDB@

Table 5: Estimated parameters by ML method for fitted distributions.

X

Observed

DBD-XII

Discrete Lomax

ZIP

Poisson

Discrete Raleigh

Geometric

0

65

63.32

61.89

64.92

27.4

11

45.98

1

14

18.19

21.01

5.82

38.08

26.83

26.76

2

10

9.29

9.65

9.52

26.47

29.55

15.57

3

6

5.49

5.24

10.38

12.26

22.23

9.06

4

4

3.52

3.17

8.48

4.26

12.49

5.28

5

2

2.39

2.06

5.55

1.18

5.42

3.07

6

2

1.69

1.42

3.02

0.27

1.85

1.79

7

2

1.23

1.02

1.41

0.05

0.5

1.04

8

1

0.92

0.76

0.58

0.01

0.11

0.61

9

1

0.7

0.58

0.21

0

0.02

0.35

10

1

0.55

0.46

0.07

0

0

0.21

11

2

2.71

2.73

0.03

0

0

0.29

χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaWbaaKqbGeqabaGaaGOmaaaaaaa@3947@ P-Values

0.532

0.352

0.0008

0.000

0.000

0.0006

Table 6: Table for goodness of fit.

Criterion

Discrete Lomax

DBD-XII

ZIP

Poisson

Discrete Raleigh

Geometric

Neg-Loglike

170.4806

168.7708

182.2449

246.21

277.778

178.7667

AIC

344.9612

343.5415

368.4897

494.42

557.556

359.5333

BIC

350.3622

351.6429

373.8907

497.1205

560.2565

362.2338

Table 7: AIC, BIC and Negative loglikelihood values for fitted distributions.

Figure 20: ECD of Counts of cysts of kidneys using steroids.
Figure 21: Overview of fitted distributions.
Figure 22: AIC, BIC and Negative loglikelihood values for fitted distributions.              

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