ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 4 Issue 7 - 2016
Garima Distribution and its Application to Model Behavioral Science Data
Rama Shanker*
Department of Statistics, Eritrea Institute of Technology, Eritrea
Received: September 30, 2016 | Published: December 09, 2016
*Corresponding author: Rama Shanker, Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea, Email:
Citation: Shanker R (2016) Garima Distribution and its Application to Model Behavioral Science Data. Biom Biostat Int J 4(7): 00116. DOI: 10.15406/bbij.2016.04.00116

Abstract

In this paper a continuous distribution named “Garima distribution” has been suggested for modeling data from behavioral science. The important properties including its shape, moments, skewness, kurtosis, hazard rate function, mean residual life function, stochastic ordering, mean deviations, order statistics, Bonferroni and Lorenz curves, entropy measure, stress-strength reliability have been discussed. The condition under which Garima distribution is over-dispersed, equi-dispersed, and under-dispersed are presented along with other one parameter continuous distributions. The estimation of its parameter has been discussed using maximum likelihood estimation and method of moments. The application of the proposed distribution has been explained using a numerical example from behavioral science and the fit has been compared with other one parameter continuous distributions.

Keywords: Lifetime distribution; Moments; Hazard rate function; Mean residual life function; Mean deviations; Order statistics; Estimation of parameter; Goodness of fit

Introduction

The modeling and analyzing lifetime data are crucial in many applied sciences including behavioral science, medicine, engineering, insurance and finance, amongst others.  There are a number of continuous distributions for modeling lifetime data such as exponential, Lindley, gamma, lognormal, and Weibull and their generalizations. The exponential, Lindley and the Weibull distributions are more popular than the gamma and the lognormal distributions because the survival functions of the gamma and the lognormal distributions cannot be expressed in closed forms and both require numerical integration. Though each of exponential and Lindley distributions has one parameter, the Lindley distribution has one advantage over the exponential distribution that the exponential distribution has constant hazard rate whereas the Lindley distribution has monotonically decreasing hazard rate.

Recently Shanker [1,2,3,4] has introduced new lifetime distributions, namely Shanker, Akash, Aradhana, and Sujatha distributions for modeling lifetime data from biomedical sciences, engineering and behavioral sciences and showed its superiority over Lindley [5] and exponential distributions. The probability density function (p.d.f.) and the cumulative distribution function (c.d.f.) of Sujatha, Aradhana, Akash, Shanker, Lindley and exponential distributions are presented in Table 1.

Distributions

Pdf

Cdf

Sujatha

f 6 ( x;θ )= θ 3 θ 2 +θ+2 ( 1+x+ x 2 ) e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGOnaaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eI7aXbGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiUde3aaWbaae qajuaibaGaaG4maaaaaKqbagaacqaH4oqCdaahaaqabKqbGeaacaaI YaaaaKqbakabgUcaRiabeI7aXjabgUcaRiaaikdaaaWaaeWaaeaaca aIXaGaey4kaSIaamiEaiabgUcaRiaadIhadaahaaqabKqbGeaacaaI YaaaaaqcfaOaayjkaiaawMcaaiaadwgadaahaaqabKqbGeaacqGHsi slcqaH4oqCcaWG4baaaaaa@566A@

F 6 ( x,θ )=1[ 1+ θx( θx+θ+2 ) θ 2 +θ+2 ] e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaaGOnaaqcfayabaWaaeWaaeaacaWG4bGaaiilaiab eI7aXbGaayjkaiaawMcaaiabg2da9iaaigdacqGHsisldaWadaqaai aaigdacqGHRaWkdaWcaaqaaiabeI7aXjaadIhadaqadaqaaiabeI7a XjaadIhacqGHRaWkcqaH4oqCcqGHRaWkcaaIYaaacaGLOaGaayzkaa aabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcqaH 4oqCcqGHRaWkcaaIYaaaaaGaay5waiaaw2faaiaadwgadaahaaqabK qbGeaacqGHsislcqaH4oqCcaWG4baaaaaa@5BAA@

Aradhana

f 5 ( x;θ )= θ 3 θ 2 +2θ+2 ( 1+x ) 2 e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGynaaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eI7aXbGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiUde3aaWbaae qajuaibaGaaG4maaaaaKqbagaacqaH4oqCdaahaaqabKqbGeaacaaI YaaaaKqbakabgUcaRiaaikdacqaH4oqCcqGHRaWkcaaIYaaaamaabm aabaGaaGymaiabgUcaRiaadIhaaiaawIcacaGLPaaadaahaaqabKqb GeaacaaIYaaaaKqbakaadwgadaahaaqabKqbGeaacqGHsislcqaH4o qCcaWG4baaaaaa@5546@

F 5 ( x;θ )=1[ 1+ θx( θx+2θ+2 ) θ 2 +2θ+2 ] e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaaGynaaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eI7aXbGaayjkaiaawMcaaiabg2da9iaaigdacqGHsisldaWadaqaai aaigdacqGHRaWkdaWcaaqaaiabeI7aXjaadIhadaqadaqaaiabeI7a XjaadIhacqGHRaWkcaaIYaGaeqiUdeNaey4kaSIaaGOmaaGaayjkai aawMcaaaqaaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4k aSIaaGOmaiabeI7aXjabgUcaRiaaikdaaaaacaGLBbGaayzxaaGaam yzamaaCaaabeqcfasaaiabgkHiTiabeI7aXjaadIhaaaaaaa@5D30@

Akash

f 4 ( x;θ )= θ 3 θ 2 +2 ( 1+ x 2 ) e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGinaaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eI7aXbGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiUde3aaWbaae qajuaibaGaaG4maaaaaKqbagaacqaH4oqCdaahaaqabKqbGeaacaaI YaaaaKqbakabgUcaRiaaikdaaaWaaeWaaeaacaaIXaGaey4kaSIaam iEamaaCaaabeqcfasaaiaaikdaaaaajuaGcaGLOaGaayzkaaGaamyz amaaCaaabeqcfasaaiabgkHiTiabeI7aXjaadIhaaaaaaa@51F1@

F 4 ( x;θ )=1[ 1+ θx( θx+2 ) θ 2 +2 ] e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaaGinaaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eI7aXbGaayjkaiaawMcaaiabg2da9iaaigdacqGHsisldaWadaqaai aaigdacqGHRaWkdaWcaaqaaiabeI7aXjaadIhadaqadaqaaiabeI7a XjaadIhacqGHRaWkcaaIYaaacaGLOaGaayzkaaaabaGaeqiUde3aaW baaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaIYaaaaaGaay5waiaa w2faaiaadwgadaahaaqabKqbGeaacqGHsislcqaH4oqCcaWG4baaaa aa@5687@

Shanker

f 3 ( x;θ )= θ 2 θ 2 +1 ( θ+x ) e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaG4maaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eI7aXbGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiUde3aaWbaae qajuaibaGaaGOmaaaaaKqbagaacqaH4oqCdaahaaqabKqbGeaacaaI YaaaaKqbakabgUcaRiaaigdaaaWaaeWaaeaacqaH4oqCcqGHRaWkca WG4baacaGLOaGaayzkaaGaamyzamaaCaaabeqcfasaaiabgkHiTiab eI7aXjaadIhaaaaaaa@514F@

F 3 ( x,θ )=1 ( θ 2 +1 )+θx θ 2 +1 e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaaG4maaqcfayabaWaaeWaaeaacaWG4bGaaiilaiab eI7aXbGaayjkaiaawMcaaiabg2da9iaaigdacqGHsisldaWcaaqaam aabmaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWk caaIXaaacaGLOaGaayzkaaGaey4kaSIaeqiUdeNaamiEaaqaaiabeI 7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGymaaaacaWG LbWaaWbaaeqajuaibaGaeyOeI0IaeqiUdeNaamiEaaaaaaa@5465@

Lindley

f 2 ( x;θ )= θ 2 θ+1 ( 1+x ) e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGOmaaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eI7aXbGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiUde3aaWbaae qajuaibaGaaGOmaaaaaKqbagaacqaH4oqCcqGHRaWkcaaIXaaaamaa bmaabaGaaGymaiabgUcaRiaadIhaaiaawIcacaGLPaaacaWGLbWaaW baaeqajuaibaGaeyOeI0IaeqiUdeNaamiEaaaaaaa@4EB9@

F 2 ( x;θ )=1[ 1+ θx θ+1 ] e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaaGOmaaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eI7aXbGaayjkaiaawMcaaiabg2da9iaaigdacqGHsisldaWadaqaai aaigdacqGHRaWkdaWcaaqaaiabeI7aXjaadIhaaeaacqaH4oqCcqGH RaWkcaaIXaaaaaGaay5waiaaw2faaiaadwgadaahaaqabKqbGeaacq GHsislcqaH4oqCcaWG4baaaaaa@4F10@

Exponential

f 1 ( x;θ )=θ e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGymaaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eI7aXbGaayjkaiaawMcaaiabg2da9iabeI7aXjaadwgadaahaaqabK qbGeaacqGHsislcqaH4oqCcaaMc8UaamiEaaaaaaa@4723@

F 1 ( x;θ )=1 e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaaGymaaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eI7aXbGaayjkaiaawMcaaiabg2da9iaaigdacqGHsislcaWGLbWaaW baaeqajuaibaGaeyOeI0IaeqiUdeNaaGPaVlaadIhaaaaaaa@46F5@

Table 1: pdf and cdf of Sujatha [4], Aradhana [3], Akash [2], Shanker[1], Lindley [5] and exponential distributions.

A New Lifetime Distribution

The probability density function (p.d.f.) of a new lifetime distribution can be introduced as

f 7 ( x;θ )= θ θ+2 ( 1+θ+θx ) e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaG4naaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eI7aXbGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiUdehabaGaeq iUdeNaey4kaSIaaGOmaaaadaqadaqaaiaaigdacqGHRaWkcqaH4oqC cqGHRaWkcqaH4oqCcaaMc8UaamiEaaGaayjkaiaawMcaaiaadwgada ahaaqabKqbGeaacqGHsislcqaH4oqCcaWG4baaaKqbakaaykW7caaM c8UaaGPaVlaaykW7caGG7aGaamiEaiabg6da+iaaicdacaGGSaGaaG PaVlaaykW7cqaH4oqCcqGH+aGpcaaIWaaaaa@6474@                          (2.1)                                          

 We would call this distribution, “Garima distribution”. This distribution can be easily expressed as a mixture of exponential ( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH4oqCaiaawIcacaGLPaaaaaa@39C3@  and gamma ( 2,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaaIYaGaaiilaiabeI7aXbGaayjkaiaawMcaaaaa@3B2F@  with mixing proportion θ+1 θ+2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqaH4oqCcqGHRaWkcaaIXaaabaGaeqiUdeNaey4kaSIaaGOmaaaa aaa@3D3B@ . We have

f 7 ( x,θ )=p g 1 ( x )+( 1p ) g 2 ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaG4naaqcfayabaWaaeWaaeaacaWG4bGaaiilaiab eI7aXbGaayjkaiaawMcaaiabg2da9iaadchacaaMc8Uaam4zamaaBa aajuaibaGaaGymaaqcfayabaWaaeWaaeaacaWG4baacaGLOaGaayzk aaGaey4kaSYaaeWaaeaacaaIXaGaeyOeI0IaamiCaaGaayjkaiaawM caaiaadEgadaWgaaqcfasaaiaaikdaaKqbagqaamaabmaabaGaamiE aaGaayjkaiaawMcaaaaa@509C@                                             (2.2)

where p= θ+1 θ+2 , g 1 ( x )=θ e θx ,and g 2 ( x )= θ 2 x e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiCai abg2da9maalaaabaGaeqiUdeNaey4kaSIaaGymaaqaaiabeI7aXjab gUcaRiaaikdaaaGaaiilaiaaykW7caaMc8Uaam4zamaaBaaajuaiba GaaGymaaqcfayabaWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaeyyp a0JaeqiUdeNaamyzamaaCaaabeqcfasaaiabgkHiTiabeI7aXjaadI haaaqcfaOaaiilaiaaykW7caaMc8Uaaeyyaiaab6gacaqGKbGaaGPa VlaadEgadaWgaaqcfasaaiaaikdaaKqbagqaamaabmaabaGaamiEaa GaayjkaiaawMcaaiabg2da9iabeI7aXnaaCaaabeqcfasaaiaaikda aaqcfaOaaGPaVlaadIhacaaMc8UaamyzamaaCaaabeqcfasaaiabgk HiTiabeI7aXjaadIhaaaaaaa@6A85@ .

The corresponding cumulative distribution function (c.d.f.) of (2.1) is given by         

F 7 ( x;θ )=1[ 1+ θx θ+2 ] e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaaG4naaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eI7aXbGaayjkaiaawMcaaiabg2da9iaaigdacqGHsisldaWadaqaai aaigdacqGHRaWkdaWcaaqaaiabeI7aXjaadIhaaeaacqaH4oqCcqGH RaWkcaaIYaaaaaGaay5waiaaw2faaiaadwgadaahaaqabKqbGeaacq GHsislcqaH4oqCcaWG4baaaaaa@4F16@ ;       x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEai abg6da+iaaicdacaGGSaGaeqiUdeNaeyOpa4JaaGimaaaa@3D6B@                               (2.3)

The graphs of the p.d.f. and the c.d.f. of Garima distributions for different values of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ are shown in Figure 1.

Figure 1: Graphs of the pdf and cdf of Garima distribution for various values of the parameter θ.

Moments and Related Measures

The r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@36ED@ the moment about origin of Garima distributon (2.1) has been obtained as

μ r = r!( θ+r+2 ) θ r ( θ+2 ) ;r=1,2,3,... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaWGYbaajuaGbeaadaahaaqabeaacWaGGBOmGika aiabg2da9maalaaabaGaamOCaiaacgcadaqadaqaaiabeI7aXjabgU caRiaadkhacqGHRaWkcaaIYaaacaGLOaGaayzkaaaabaGaeqiUde3a aWbaaeqajuaibaGaamOCaaaajuaGdaqadaqaaiabeI7aXjabgUcaRi aaikdaaiaawIcacaGLPaaaaaGaaGPaVlaaykW7caaMc8Uaai4oaiaa dkhacqGH9aqpcaaIXaGaaiilaiaaikdacaGGSaGaaG4maiaacYcaca GGUaGaaiOlaiaac6caaaa@5C9F@

and so the first four moments about origin as

μ 1 = θ+3 θ( θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIXaaajuaGbeaadaahaaqabeaacWaGGBOmGika aiabg2da9maalaaabaGaeqiUdeNaey4kaSIaaG4maaqaaiabeI7aXn aabmaabaGaeqiUdeNaey4kaSIaaGOmaaGaayjkaiaawMcaaaaaaaa@47DA@ ,         μ 2 = 2( θ+4 ) θ 2 ( θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaadaahaaqabeaacWaGGBOmGika aiabg2da9maalaaabaGaaGOmamaabmaabaGaeqiUdeNaey4kaSIaaG inaaGaayjkaiaawMcaaaqaaiabeI7aXnaaCaaabeqcfasaaiaaikda aaqcfa4aaeWaaeaacqaH4oqCcqGHRaWkcaaIYaaacaGLOaGaayzkaa aaaaaa@4BBB@ ,          μ 3 = 6( θ+5 ) θ 3 ( θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIZaaajuaGbeaadaahaaqabeaacWaGGBOmGika aiabg2da9maalaaabaGaaGOnamaabmaabaGaeqiUdeNaey4kaSIaaG ynaaGaayjkaiaawMcaaaqaaiabeI7aXnaaCaaabeqcfasaaiaaioda aaqcfa4aaeWaaeaacqaH4oqCcqGHRaWkcaaIYaaacaGLOaGaayzkaa aaaaaa@4BC2@ ,         μ 4 = 24( θ+6 ) θ 4 ( θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaI0aaajuaGbeaadaahaaqabeaacWaGGBOmGika aiabg2da9maalaaabaGaaGOmaiaaisdadaqadaqaaiabeI7aXjabgU caRiaaiAdaaiaawIcacaGLPaaaaeaacqaH4oqCdaahaaqabKqbGeaa caaI0aaaaKqbaoaabmaabaGaeqiUdeNaey4kaSIaaGOmaaGaayjkai aawMcaaaaaaaa@4C7F@

Using the relationship between central moments and the moments about origin, the central moments of Garima distribution are obtained as

μ 2 = θ 2 +6θ+7 θ 2 ( θ+2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpdaWcaaqaaiabeI7a XnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGOnaiabeI7aXj abgUcaRiaaiEdaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqb aoaabmaabaGaeqiUdeNaey4kaSIaaGOmaaGaayjkaiaawMcaamaaCa aabeqcfasaaiaaikdaaaaaaaaa@4C6D@

μ 3 = 2( θ 3 +9 θ 2 +21θ+15 ) θ 3 ( θ+2 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIZaaajuaGbeaacqGH9aqpdaWcaaqaaiaaikda daqadaqaaiabeI7aXnaaCaaabeqcfasaaiaaiodaaaqcfaOaey4kaS IaaGyoaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIa aGOmaiaaigdacqaH4oqCcqGHRaWkcaaIXaGaaGynaaGaayjkaiaawM caaaqaaiabeI7aXnaaCaaabeqcfasaaiaaiodaaaqcfa4aaeWaaeaa cqaH4oqCcqGHRaWkcaaIYaaacaGLOaGaayzkaaWaaWbaaeqajuaiba GaaG4maaaaaaaaaa@551B@

μ 4 = 3( 3 θ 4 +36 θ 3 +134 θ 2 +204θ+111 ) θ 4 ( θ+2 ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaI0aaajuaGbeaacqGH9aqpdaWcaaqaaiaaioda daqadaqaaiaaiodacqaH4oqCdaahaaqabKqbGeaacaaI0aaaaKqbak abgUcaRiaaiodacaaI2aGaeqiUde3aaWbaaeqajuaibaGaaG4maaaa juaGcqGHRaWkcaaIXaGaaG4maiaaisdacqaH4oqCdaahaaqabKqbGe aacaaIYaaaaKqbakabgUcaRiaaikdacaaIWaGaaGinaiabeI7aXjab gUcaRiaaigdacaaIXaGaaGymaaGaayjkaiaawMcaaaqaaiabeI7aXn aaCaaabeqcfasaaiaaisdaaaqcfa4aaeWaaeaacqaH4oqCcqGHRaWk caaIYaaacaGLOaGaayzkaaWaaWbaaeqajuaibaGaaGinaaaaaaaaaa@5E74@

Thus the coefficient of variation ( C.V ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWGdbGaaiOlaiaadAfaaiaawIcacaGLPaaaaaa@3A62@ , coefficient of skewness ( β 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aadaGcaaqaaiabek7aInaaBaaajuaibaGaaGymaaqcfayabaaabeaa aiaawIcacaGLPaaaaaa@3B56@ , coefficient of kurtosis ( β 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHYoGydaWgaaqcfasaaiaaikdaaKqbagqaaaGaayjkaiaawMca aaaa@3B47@ and index of dispersion ( γ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHZoWzaiaawIcacaGLPaaaaaa@39B4@  of Garima distribution are obtained as

C.V= σ μ 1 = θ 2 +6θ+7 θ+3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4qai aac6cacaWGwbGaeyypa0ZaaSaaaeaacqaHdpWCaeaacqaH8oqBdaWg aaqcfasaaiaaigdaaKqbagqaamaaCaaabeqaaiadacUHYaIOaaaaai abg2da9maalaaabaWaaOaaaeaacqaH4oqCdaahaaqabKqbGeaacaaI YaaaaKqbakabgUcaRiaaiAdacqaH4oqCcqGHRaWkcaaI3aaabeaaae aacqaH4oqCcqGHRaWkcaaIZaaaaaaa@4ED0@

β 1 = μ 3 μ 2 3/2 = 2( θ 3 +9 θ 2 +21θ+15 ) ( θ 2 +6θ+7 ) 3/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaae aacqaHYoGydaWgaaqcfasaaiaaigdaaKqbagqaaaqabaGaeyypa0Za aSaaaeaacqaH8oqBdaWgaaqcfasaaiaaiodaaKqbagqaaaqaaiabeY 7aTnaaBaaajuaibaGaaGOmaaqcfayabaWaaWbaaeqajuaibaGaaG4m aiaac+cacaaIYaaaaaaajuaGcqGH9aqpdaWcaaqaaiaaikdadaqada qaaiabeI7aXnaaCaaabeqcfasaaiaaiodaaaqcfaOaey4kaSIaaGyo aiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGOmai aaigdacqaH4oqCcqGHRaWkcaaIXaGaaGynaaGaayjkaiaawMcaaaqa amaabmaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRa WkcaaI2aGaeqiUdeNaey4kaSIaaG4naaGaayjkaiaawMcaamaaCaaa beqcfasaaiaaiodacaGGVaGaaGOmaaaaaaaaaa@62E8@

β 2 = μ 4 μ 2 2 = 3( 3 θ 4 +36 θ 3 +134 θ 2 +204θ+111 ) ( θ 2 +6θ+7 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpdaWcaaqaaiabeY7a TnaaBaaajuaibaGaaGinaaqcfayabaaabaGaeqiVd02aaSbaaKqbGe aacaaIYaaajuaGbeaadaahaaqabKqbGeaacaaIYaaaaaaajuaGcqGH 9aqpdaWcaaqaaiaaiodadaqadaqaaiaaiodacqaH4oqCdaahaaqabK qbGeaacaaI0aaaaKqbakabgUcaRiaaiodacaaI2aGaeqiUde3aaWba aeqajuaibaGaaG4maaaajuaGcqGHRaWkcaaIXaGaaG4maiaaisdacq aH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaaikdacaaI WaGaaGinaiabeI7aXjabgUcaRiaaigdacaaIXaGaaGymaaGaayjkai aawMcaaaqaamaabmaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaa juaGcqGHRaWkcaaI2aGaeqiUdeNaey4kaSIaaG4naaGaayjkaiaawM caamaaCaaabeqcfasaaiaaikdaaaaaaaaa@6950@

γ= σ 2 μ 1 = θ 2 +6θ+7 θ( θ+2 )( θ+3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC Maeyypa0ZaaSaaaeaacqaHdpWCdaahaaqabKqbGeaacaaIYaaaaaqc fayaaiabeY7aTnaaBaaajuaibaGaaGymaaqcfayabaWaaWbaaeqaba Gamai4gkdiIcaaaaGaeyypa0ZaaSaaaeaacqaH4oqCdaahaaqabKqb GeaacaaIYaaaaKqbakabgUcaRiaaiAdacqaH4oqCcqGHRaWkcaaI3a aabaGaeqiUde3aaeWaaeaacqaH4oqCcqGHRaWkcaaIYaaacaGLOaGa ayzkaaWaaeWaaeaacqaH4oqCcqGHRaWkcaaIZaaacaGLOaGaayzkaa aaaaaa@57C8@

The condition under which Garima distribution is over-dispersed ( μ< σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH8oqBcqGH8aapcqaHdpWCdaahaaqabKqbGeaacaaIYaaaaaqc faOaayjkaiaawMcaaaaa@3E24@ , equi-dispersed ( μ= σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH8oqBcqGH9aqpcqaHdpWCdaahaaqabKqbGeaacaaIYaaaaaqc faOaayjkaiaawMcaaaaa@3E26@ and under-dispersed ( μ> σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH8oqBcqGH+aGpcqaHdpWCdaahaaqabKqbGeaacaaIYaaaaaqc faOaayjkaiaawMcaaaaa@3E28@ are presented in Table 2 along with other lifetime distributions.

Lifetime
Distributions

Over-Dispersion ( μ< σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH8oqBcqGH8aapcqaHdpWCdaahaaqabKqbGeaacaaIYaaaaaqc faOaayjkaiaawMcaaaaa@3E24@

Equi-Dispersion
( μ= σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH8oqBcqGH9aqpcqaHdpWCdaahaaqabKqbGeaacaaIYaaaaaqc faOaayjkaiaawMcaaaaa@3E26@

Under-Dispersion
( μ> σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH8oqBcqGH+aGpcqaHdpWCdaahaaqabKqbGeaacaaIYaaaaaqc faOaayjkaiaawMcaaaaa@3E28@

Garima

θ<1.164247938 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyipaWJaaGymaiaac6cacaaIXaGaaGOnaiaaisdacaaIYaGaaGin aiaaiEdacaaI5aGaaG4maiaaiIdaaaa@4161@

θ=1.164247938 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaaGymaiaac6cacaaIXaGaaGOnaiaaisdacaaIYaGaaGin aiaaiEdacaaI5aGaaG4maiaaiIdaaaa@4163@

θ>1.164247938 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyOpa4JaaGymaiaac6cacaaIXaGaaGOnaiaaisdacaaIYaGaaGin aiaaiEdacaaI5aGaaG4maiaaiIdaaaa@4165@

Sujatha

θ<1.364271174 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyipaWJaaGymaiaac6cacaaIZaGaaGOnaiaaisdacaaIYaGaaG4n aiaaigdacaaIXaGaaG4naiaaisdaaaa@4158@

θ=1.364271174 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaaGymaiaac6cacaaIZaGaaGOnaiaaisdacaaIYaGaaG4n aiaaigdacaaIXaGaaG4naiaaisdaaaa@415A@

θ>1.364271174 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyOpa4JaaGymaiaac6cacaaIZaGaaGOnaiaaisdacaaIYaGaaG4n aiaaigdacaaIXaGaaG4naiaaisdaaaa@415C@

Aradhana

θ<1.283826505 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyipaWJaaGymaiaac6cacaaIYaGaaGioaiaaiodacaaI4aGaaGOm aiaaiAdacaaI1aGaaGimaiaaiwdaaaa@415C@

θ=1.283826505 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaaGymaiaac6cacaaIYaGaaGioaiaaiodacaaI4aGaaGOm aiaaiAdacaaI1aGaaGimaiaaiwdaaaa@415E@

θ>1.283826505 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyOpa4JaaGymaiaac6cacaaIYaGaaGioaiaaiodacaaI4aGaaGOm aiaaiAdacaaI1aGaaGimaiaaiwdaaaa@4160@

Akash

θ<1.515400063 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyipaWJaaGymaiaac6cacaaI1aGaaGymaiaaiwdacaaI0aGaaGim aiaaicdacaaIWaGaaGOnaiaaiodaaaa@414D@

θ=1.515400063 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaaGymaiaac6cacaaI1aGaaGymaiaaiwdacaaI0aGaaGim aiaaicdacaaIWaGaaGOnaiaaiodaaaa@414F@

θ>1.515400063 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyOpa4JaaGymaiaac6cacaaI1aGaaGymaiaaiwdacaaI0aGaaGim aiaaicdacaaIWaGaaGOnaiaaiodaaaa@4151@

Shanker

θ<1.171535555 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyipaWJaaGymaiaac6cacaaIXaGaaG4naiaaigdacaaI1aGaaG4m aiaaiwdacaaI1aGaaGynaiaaiwdaaaa@415A@

θ=1.171535555 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaaGymaiaac6cacaaIXaGaaG4naiaaigdacaaI1aGaaG4m aiaaiwdacaaI1aGaaGynaiaaiwdaaaa@415C@

θ>1.171535555 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyOpa4JaaGymaiaac6cacaaIXaGaaG4naiaaigdacaaI1aGaaG4m aiaaiwdacaaI1aGaaGynaiaaiwdaaaa@415E@

Lindley

θ<1.170086487 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyipaWJaaGymaiaac6cacaaIXaGaaG4naiaaicdacaaIWaGaaGio aiaaiAdacaaI0aGaaGioaiaaiEdaaaa@415E@

θ=1.170086487 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaaGymaiaac6cacaaIXaGaaG4naiaaicdacaaIWaGaaGio aiaaiAdacaaI0aGaaGioaiaaiEdaaaa@4160@

θ>1.170086487 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyOpa4JaaGymaiaac6cacaaIXaGaaG4naiaaicdacaaIWaGaaGio aiaaiAdacaaI0aGaaGioaiaaiEdaaaa@4162@

Exponential

θ<1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyipaWJaaGymaaaa@39F9@

θ=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaaGymaaaa@39FB@

θ>1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyOpa4JaaGymaaaa@39FD@

Table 2: Over-dispersion, equi-dispersion and under-dispersion of Garima, Sujatha [4], Aradhana [3], Akash [2], Shanker [1], Lindley [5], and exponential distributions for varying values of their parameter θ.

Generating Functions

The moment generating function ( M X ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWGnbWaaSbaaKqbGeaacaWGybaajuaGbeaadaqadaqaaiaadsha aiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3D1B@ , characteristic function ( φ X ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHgpGAdaWgaaqcfasaaiaadIfaaKqbagqaamaabmaabaGaamiD aaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa@3E06@ , and cumulant generating function ( K X ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWGlbWaaSbaaKqbGeaacaWGybaajuaGbeaadaqadaqaaiaadsha aiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3D19@ of Garima distribution (1.3) are given by

M X ( t )=( 1 ( θ+1 )t θ 2 +2θ ) ( 1 t θ ) 2 ,| t θ |1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytam aaBaaajuaibaGaamiwaaqcfayabaWaaeWaaeaacaWG0baacaGLOaGa ayzkaaGaeyypa0ZaaeWaaeaacaaIXaGaeyOeI0YaaSaaaeaadaqada qaaiabeI7aXjabgUcaRiaaigdaaiaawIcacaGLPaaacaWG0baabaGa eqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaIYaGaeq iUdehaaaGaayjkaiaawMcaamaabmaabaGaaGymaiabgkHiTmaalaaa baGaamiDaaqaaiabeI7aXbaaaiaawIcacaGLPaaadaahaaqabKqbGe aacqGHsislcaaIYaaaaKqbakaaykW7caaMc8UaaGPaVlaacYcadaab daqaamaalaaabaGaamiDaaqaaiabeI7aXbaaaiaawEa7caGLiWoacq GHKjYOcaaIXaaaaa@626B@

                 

φ X ( t )=( 1 ( θ+1 )it θ 2 +2θ ) ( 1 it θ ) 2 ,i= 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOXdO 2aaSbaaKqbGeaacaWGybaajuaGbeaadaqadaqaaiaadshaaiaawIca caGLPaaacqGH9aqpdaqadaqaaiaaigdacqGHsisldaWcaaqaamaabm aabaGaeqiUdeNaey4kaSIaaGymaaGaayjkaiaawMcaaiaadMgacaWG 0baabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkca aIYaGaeqiUdehaaaGaayjkaiaawMcaamaabmaabaGaaGymaiabgkHi TmaalaaabaGaamyAaiaadshaaeaacqaH4oqCaaaacaGLOaGaayzkaa WaaWbaaeqajuaibaGaeyOeI0IaaGOmaaaajuaGcaaMc8UaaGPaVlaa ykW7caGGSaGaamyAaiabg2da9maakaaabaGaeyOeI0IaaGymaaqaba aaaa@608D@

K X ( t )=log( 1 ( θ+1 )it θ 2 +2θ )2log( 1 it θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4sam aaBaaajuaibaGaamiwaaqcfayabaWaaeWaaeaacaWG0baacaGLOaGa ayzkaaGaeyypa0JaciiBaiaac+gacaGGNbWaaeWaaeaacaaIXaGaey OeI0YaaSaaaeaadaqadaqaaiabeI7aXjabgUcaRiaaigdaaiaawIca caGLPaaacaWGPbGaamiDaaqaaiabeI7aXnaaCaaabeqcfasaaiaaik daaaqcfaOaey4kaSIaaGOmaiabeI7aXbaaaiaawIcacaGLPaaacqGH sislcaaIYaGaciiBaiaac+gacaGGNbWaaeWaaeaacaaIXaGaeyOeI0 YaaSaaaeaacaWGPbGaamiDaaqaaiabeI7aXbaaaiaawIcacaGLPaaa aaa@5B65@

Using the expansion log( 1x )= r=0 x r r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaciiBai aac+gacaGGNbWaaeWaaeaacaaIXaGaeyOeI0IaamiEaaGaayjkaiaa wMcaaiabg2da9iabgkHiTmaaqahabaWaaSaaaeaacaWG4bWaaWbaae qajuaibaGaamOCaaaaaKqbagaacaWGYbaaaaqcfasaaiaadkhacqGH 9aqpcaaIWaaabaGaeyOhIukajuaGcqGHris5aaaa@4A5C@ , we get

K X ( t )= r=0 ( θ+1 θ 2 +2θ ) r ( it ) r r +2 r=0 ( it θ ) r r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4sam aaBaaajuaibaGaamiwaaqcfayabaWaaeWaaeaacaWG0baacaGLOaGa ayzkaaGaeyypa0JaeyOeI0YaaabCaeaadaqadaqaamaalaaabaGaeq iUdeNaey4kaSIaaGymaaqaaiabeI7aXnaaCaaabeqcfasaaiaaikda aaqcfaOaey4kaSIaaGOmaiabeI7aXbaaaiaawIcacaGLPaaadaahaa qabKqbGeaacaWGYbaaaaqaaiaadkhacqGH9aqpcaaIWaaabaGaeyOh IukajuaGcqGHris5amaalaaabaWaaeWaaeaacaWGPbGaaGPaVlaads haaiaawIcacaGLPaaadaahaaqabKqbGeaacaWGYbaaaaqcfayaaiaa dkhaaaGaey4kaSIaaGOmamaaqahabaWaaSaaaeaadaqadaqaamaali aabaGaamyAaiaadshaaeaacqaH4oqCaaaacaGLOaGaayzkaaWaaWba aeqajuaibaGaamOCaaaaaKqbagaacaWGYbaaaaqcfasaaiaadkhacq GH9aqpcaaIWaaabaGaeyOhIukajuaGcqGHris5aaaa@69D1@

=2 r=0 1 θ r ( it ) r r r=0 ( θ+1 θ 2 +2θ ) r ( it ) r r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 JaaGOmamaaqahabaWaaSaaaeaacaaIXaaabaGaeqiUde3aaWbaaeqa juaibaGaamOCaaaaaaqcfa4aaSaaaeaadaqadaqaaiaadMgacaWG0b aacaGLOaGaayzkaaWaaWbaaeqajuaibaGaamOCaaaaaKqbagaacaWG YbaaaaqcfasaaiaadkhacqGH9aqpcaaIWaaabaGaeyOhIukajuaGcq GHris5aiabgkHiTmaaqahabaWaaeWaaeaadaWcaaqaaiabeI7aXjab gUcaRiaaigdaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbak abgUcaRiaaikdacqaH4oqCaaaacaGLOaGaayzkaaWaaWbaaeqajuai baGaamOCaaaaaeaacaWGYbGaeyypa0JaaGimaaqaaiabg6HiLcqcfa OaeyyeIuoadaWcaaqaamaabmaabaGaamyAaiaadshaaiaawIcacaGL PaaadaahaaqabKqbGeaacaWGYbaaaaqcfayaaiaadkhaaaaaaa@64E6@

=2 r=0 ( r1 )! θ r ( it ) r r! r=0 ( θ+1 θ 2 +2θ ) r ( r1 )! ( it ) r r! MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 JaaGOmamaaqahabaWaaSaaaeaadaqadaqaaiaadkhacqGHsislcaaI XaaacaGLOaGaayzkaaGaaiyiaaqaaiabeI7aXnaaCaaabeqcfasaai aadkhaaaaaaKqbaoaalaaabaWaaeWaaeaacaWGPbGaamiDaaGaayjk aiaawMcaamaaCaaabeqcfasaaiaadkhaaaaajuaGbaGaamOCaiaacg caaaaajuaibaGaamOCaiabg2da9iaaicdaaeaacqGHEisPaKqbakab ggHiLdGaeyOeI0YaaabCaeaadaqadaqaamaalaaabaGaeqiUdeNaey 4kaSIaaGymaaqaaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOa ey4kaSIaaGOmaiabeI7aXbaaaiaawIcacaGLPaaadaahaaqabKqbGe aacaWGYbaaaKqbaoaabmaabaGaamOCaiabgkHiTiaaigdaaiaawIca caGLPaaacaGGHaaajuaibaGaamOCaiabg2da9iaaicdaaeaacqGHEi sPaKqbakabggHiLdWaaSaaaeaadaqadaqaaiaadMgacaWG0baacaGL OaGaayzkaaWaaWbaaeqajuaibaGaamOCaaaaaKqbagaacaWGYbGaai yiaaaaaaa@6FCB@

Thus the r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@36ED@ th cumulant of Garima distribution is given by

K r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4sam aaBaaajuaibaGaamOCaaqcfayabaaaaa@3928@ = coefficient of ( it ) r r! MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aadaqadaqaaiaadMgacaWG0baacaGLOaGaayzkaaWaaWbaaeqajuai baGaamOCaaaaaKqbagaacaWGYbGaaiyiaaaaaaa@3D75@ in K X ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4sam aaBaaajuaibaGaamiwaaqcfayabaWaaeWaaeaacaWG0baacaGLOaGa ayzkaaaaaa@3B90@                      

= 2( r1 )! θ r ( r1 )! ( θ+1 ) r ( θ 2 +2θ ) r ;r=1,2,3,... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacaaIYaWaaeWaaeaacaWGYbGaeyOeI0IaaGymaaGaayjk aiaawMcaaiaacgcaaeaacqaH4oqCdaahaaqabKqbGeaacaWGYbaaaa aajuaGcqGHsisldaWcaaqaamaabmaabaGaamOCaiabgkHiTiaaigda aiaawIcacaGLPaaacaGGHaWaaeWaaeaacqaH4oqCcqGHRaWkcaaIXa aacaGLOaGaayzkaaWaaWbaaeqajuaibaGaamOCaaaaaKqbagaadaqa daqaaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaG OmaiabeI7aXbGaayjkaiaawMcaamaaCaaabeqcfasaaiaadkhaaaaa aKqbakaaykW7caaMc8Uaai4oaiaadkhacqGH9aqpcaaIXaGaaiilai aaikdacaGGSaGaaG4maiaacYcacaGGUaGaaiOlaiaac6caaaa@6357@

This gives

μ 1 = K 1 = θ+3 θ( θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIXaaajuaGbeaadaahaaqabeaacWaGGBOmGika aiabg2da9iaadUeadaWgaaqcfasaaiaaigdaaKqbagqaaiabg2da9m aalaaabaGaeqiUdeNaey4kaSIaaG4maaqaaiabeI7aXnaabmaabaGa eqiUdeNaey4kaSIaaGOmaaGaayjkaiaawMcaaaaaaaa@4B48@

μ 2 = K 2 = θ 2 +6θ+7 θ 2 ( θ+2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpcaWGlbWaaSbaaKqb GeaacaaIYaaajuaGbeaacqGH9aqpdaWcaaqaaiabeI7aXnaaCaaabe qcfasaaiaaikdaaaqcfaOaey4kaSIaaGOnaiabeI7aXjabgUcaRiaa iEdaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbaoaabmaaba GaeqiUdeNaey4kaSIaaGOmaaGaayjkaiaawMcaamaaCaaabeqcfasa aiaaikdaaaaaaaaa@4FDC@

μ 3 = K 3 = 2( θ 3 +9 θ 2 +21θ+15 ) θ 3 ( θ+2 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIZaaajuaGbeaacqGH9aqpcaWGlbWaaSbaaKqb GeaacaaIZaaajuaGbeaacqGH9aqpdaWcaaqaaiaaikdadaqadaqaai abeI7aXnaaCaaabeqcfasaaiaaiodaaaqcfaOaey4kaSIaaGyoaiab eI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGOmaiaaig dacqaH4oqCcqGHRaWkcaaIXaGaaGynaaGaayjkaiaawMcaaaqaaiab eI7aXnaaCaaabeqcfasaaiaaiodaaaqcfa4aaeWaaeaacqaH4oqCcq GHRaWkcaaIYaaacaGLOaGaayzkaaWaaWbaaeqajuaibaGaaG4maaaa aaaaaa@588B@

μ 4 = K 4 +3 K 2 2 = 3( 3 θ 4 +36 θ 3 +134 θ 2 +204θ+111 ) θ 4 ( θ+2 ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaI0aaajuaGbeaacqGH9aqpcaWGlbWaaSbaaKqb GeaacaaI0aaajuaGbeaacqGHRaWkcaaIZaGaam4samaaBaaajuaiba GaaGOmaaqcfayabaWaaWbaaeqajuaibaGaaGOmaaaajuaGcqGH9aqp daWcaaqaaiaaiodadaqadaqaaiaaiodacqaH4oqCdaahaaqabKqbGe aacaaI0aaaaKqbakabgUcaRiaaiodacaaI2aGaeqiUde3aaWbaaeqa juaibaGaaG4maaaajuaGcqGHRaWkcaaIXaGaaG4maiaaisdacqaH4o qCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaaikdacaaIWaGa aGinaiabeI7aXjabgUcaRiaaigdacaaIXaGaaGymaaGaayjkaiaawM caaaqaaiabeI7aXnaaCaaabeqcfasaaiaaisdaaaqcfa4aaeWaaeaa cqaH4oqCcqGHRaWkcaaIYaaacaGLOaGaayzkaaWaaWbaaeqajuaiba GaaGinaaaaaaaaaa@6787@

Which the same are as obtained earlier.

Hazard Rate Function and Mean Residual Life Function

Let X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwaa aa@3761@ be a continuous random variable with pdf f( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamiEaaGaayjkaiaawMcaaaaa@39F5@ and cdf F( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aabmaabaGaamiEaaGaayjkaiaawMcaaaaa@39D5@ . The hazard rate function (also known as the failure rate function) and the mean residual life function of X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwaa aa@3761@ are respectively defined as

h( x )= lim Δx0 P( X<x+Δx|X>x ) Δx = f 7 ( x;θ ) 1 F 7 ( x;θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9maaxababaGaciiB aiaacMgacaGGTbaajuaibaGaeyiLdqKaamiEaiabgkziUkaaicdaaK qbagqaamaalaaabaGaamiuamaabmaabaWaaqGaaeaacaWGybGaeyip aWJaamiEaiabgUcaRiabgs5aejaadIhaaiaawIa7aiaadIfacqGH+a GpcaWG4baacaGLOaGaayzkaaaabaGaeyiLdqKaamiEaaaacqGH9aqp daWcaaqaaiaadAgadaWgaaqcfasaaiaaiEdaaKqbagqaamaabmaaba GaamiEaiaacUdacqaH4oqCaiaawIcacaGLPaaaaeaacaaIXaGaeyOe I0IaamOramaaBaaajuaibaGaaG4naaqcfayabaWaaeWaaeaacaWG4b Gaai4oaiabeI7aXbGaayjkaiaawMcaaaaaaaa@64D6@                                         (5.1)

and  m( x )=E[ Xx|X>x ]= 1 1 F 7 ( x;θ ) x [ 1 F 7 ( t;θ ) ] dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaeiiai aad2gadaqadaqaaiaadIhaaiaawIcacaGLPaaacqGH9aqpcaWGfbWa amWaaeaadaabcaqaaiaadIfacqGHsislcaWG4baacaGLiWoacaWGyb GaeyOpa4JaamiEaaGaay5waiaaw2faaiaaysW7cqGH9aqpcaaMe8+a aSaaaeaacaaIXaaabaGaaGymaiabgkHiTiaadAeadaWgaaqcfasaai aaiEdaaKqbagqaamaabmaabaGaamiEaiaacUdacqaH4oqCaiaawIca caGLPaaaaaWaa8qmaeaadaWadaqaaiaaigdacqGHsislcaWGgbWaaS baaKqbGeaacaaI3aaajuaGbeaadaqadaqaaiaadshacaGG7aGaeqiU dehacaGLOaGaayzkaaaacaGLBbGaayzxaaaajuaibaGaamiEaaqaai abg6HiLcqcfaOaey4kIipacaaMe8UaaGPaVlaadsgacaWG0baaaa@68C6@                              (5.2)

The hazard rate function, h( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aabmaabaGaamiEaaGaayjkaiaawMcaaaaa@39F7@ and the mean residual life function, m( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyBam aabmaabaGaamiEaaGaayjkaiaawMcaaaaa@39FC@  of Garima distribution are given by

h( x )= θ( 1+θ+x ) θx+( θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiU de3aaeWaaeaacaaIXaGaey4kaSIaeqiUdeNaey4kaSIaamiEaaGaay jkaiaawMcaaaqaaiabeI7aXjaadIhacqGHRaWkdaqadaqaaiabeI7a XjabgUcaRiaaikdaaiaawIcacaGLPaaaaaaaaa@4BF0@                                                                   (5.3)

and                    m( x )= θx+θ+3 θ( θx+θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyBam aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiU deNaamiEaiabgUcaRiabeI7aXjabgUcaRiaaiodaaeaacqaH4oqCda qadaqaaiabeI7aXjaadIhacqGHRaWkcqaH4oqCcqGHRaWkcaaIYaaa caGLOaGaayzkaaaaaaaa@4C24@                                                                 (5.4)

It can be easily verified that h( 0 )= θ( θ+1 ) θ+2 =f( 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aabmaabaGaaGimaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiU de3aaeWaaeaacqaH4oqCcqGHRaWkcaaIXaaacaGLOaGaayzkaaaaba GaeqiUdeNaey4kaSIaaGOmaaaacqGH9aqpcaWGMbWaaeWaaeaacaaI WaaacaGLOaGaayzkaaaaaa@48E4@ and m( 0 )= θ+3 θ( θ+2 ) = μ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyBam aabmaabaGaaGimaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiU deNaey4kaSIaaG4maaqaaiabeI7aXnaabmaabaGaeqiUdeNaey4kaS IaaGOmaaGaayjkaiaawMcaaaaacqGH9aqpcqaH8oqBdaWgaaqcfasa aiaaigdaaKqbagqaamaaCaaabeqaaiadacUHYaIOaaaaaa@4C15@ .It is also obvious from the graphs of h( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aabmaabaGaamiEaaGaayjkaiaawMcaaaaa@39F7@  and m( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyBam aabmaabaGaamiEaaGaayjkaiaawMcaaaaa@39FC@ that h( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aabmaabaGaamiEaaGaayjkaiaawMcaaaaa@39F7@  is an increasing or decreasing function of x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F3@ , and θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ , where as m( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyBam aabmaabaGaamiEaaGaayjkaiaawMcaaaaa@39FC@ is a decreasing function of x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F3@ , and θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ . The graph of the hazard rate function and mean residual life function of Garima distribution are shown in Figures 2 & 3.         

Figure 2: Graph of hazard rate function of Garima distribution for different values of parameter θ.

Figure 3: Graph of mean residual life function of Garima distribution for different values of parameter θ.

Stochastic Orderings

Stochastic ordering of positive continuous random variables is an important tool for judging their comparative behavior. A random variable X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwaa aa@3761@ is said to be smaller than a random variable Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamywaa aa@3762@ in the

  1. stochastic order ( X st Y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWGybGaeyizIm6aaSbaaKqbGeaacaWGZbGaamiDaaqcfayabaGa amywaaGaayjkaiaawMcaaaaa@3E4B@ if F X ( x ) F Y ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaamiwaaqcfayabaWaaeWaaeaacaWG4baacaGLOaGa ayzkaaGaeyyzImRaamOramaaBaaajuaibaGaamywaaqcfayabaWaae WaaeaacaWG4baacaGLOaGaayzkaaaaaa@4261@ for all x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEaa aa@3781@
  2. hazard rate order ( X hr Y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWGybGaeyizIm6aaSbaaKqbGeaacaWGObGaamOCaaqcfayabaGa amywaaGaayjkaiaawMcaaaaa@3E3E@ if h X ( x ) h Y ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aaBaaajuaibaGaamiwaaqcfayabaWaaeWaaeaacaWG4baacaGLOaGa ayzkaaGaeyyzImRaamiAamaaBaaajuaibaGaamywaaqcfayabaWaae WaaeaacaWG4baacaGLOaGaayzkaaaaaa@42A5@  for all
  3. mean residual life order ( X mrl Y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWGybGaeyizIm6aaSbaaKqbGeaacaWGTbGaamOCaiaadYgaaKqb agqaaiaadMfaaiaawIcacaGLPaaaaaa@3F34@ if m X ( x ) m Y ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyBam aaBaaajuaibaGaamiwaaqcfayabaWaaeWaaeaacaWG4baacaGLOaGa ayzkaaGaeyizImQaamyBamaaBaaajuaibaGaamywaaqcfayabaWaae WaaeaacaWG4baacaGLOaGaayzkaaaaaa@429E@ for all x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F3@
  4. likelihood ratio order ( X lr Y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWGybGaeyizIm6aaSbaaKqbGeaacaWGSbGaamOCaaqcfayabaGa amywaaGaayjkaiaawMcaaaaa@3E42@ if f X ( x ) f Y ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGMbWaaSbaaKqbGeaacaWGybaajuaGbeaadaqadaqaaiaadIha aiaawIcacaGLPaaaaeaacaWGMbWaaSbaaKqbGeaacaWGzbaajuaGbe aadaqadaqaaiaadIhaaiaawIcacaGLPaaaaaaaaa@40EB@  decreases in x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F3@ .

The following results due to Shaked & Shanthikumar [6] are well known for establishing stochastic ordering of distributions

X lr YX hr YX mrl Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaajuaibaGaamiBaiaadkhaaKqbagqaaiaadMfacqGH shI3caWGybGaeyizIm6aaSbaaKqbGeaacaWGObGaamOCaaqcfayaba GaamywaiabgkDiElaadIfacqGHKjYOdaWgaaqcfasaaiaad2gacaWG YbGaamiBaaqcfayabaGaamywaaaa@4ECB@                                            (6.1)

X st Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaCbeae aacqGHthY3aeaacaWGybGaeyizIm6aaSbaaKqbGeaacaWGZbGaamiD aaqcfayabaGaamywaaqabaaaaa@3F4F@

The Garima distribution is ordered with respect to the strongest ‘likelihood ratio’ ordering as shown in the following theorem:

  1. Theorem: Let X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwaa aa@3761@ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaGae8hpI4 haaa@3761@  Garima distributon ( θ 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH4oqCdaWgaaqcfasaaiaaigdaaKqbagqaaaGaayjkaiaawMca aaaa@3B5B@  and Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamywaa aa@3762@ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaGae8hpI4 haaa@3761@  Garima distribution ( θ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH4oqCdaWgaaqcfasaaiaaikdaaKqbagqaaaGaayjkaiaawMca aaaa@3B5C@ . If θ 1 θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde 3aaSbaaKqbGeaacaaIXaaajuaGbeaacqGHLjYScqaH4oqCdaWgaaqc fasaaiaaikdaaKqbagqaaaaa@3EE7@ , then X lr Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaajuaibaGaamiBaiaadkhaaKqbagqaaiaadMfaaaa@3CB9@ and hence X hr Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaajuaibaGaamiAaiaadkhaaKqbagqaaiaadMfaaaa@3CB5@ , X mrl Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaajuaibaGaamyBaiaadkhacaWGSbaajuaGbeaacaWG zbaaaa@3DAB@ and X st Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaajuaibaGaam4CaiaadshaaKqbagqaaiaadMfaaaa@3CC2@ .
  2. Proof: We have

f X ( x ) f Y ( x ) = θ 1 ( θ 2 +2 ) θ 2 ( θ 1 +2 ) ( 1+ θ 1 + θ 1 x 1+ θ 2 + θ 2 x ) e ( θ 1 θ 2 )x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSqaae aacaWGMbWaaSbaaKqbGeaacaWGybaajuaGbeaadaqadaqaaiaadIha aiaawIcacaGLPaaaaeaacaWGMbWaaSbaaKqbGeaacaWGzbaajuaGbe aadaqadaqaaiaadIhaaiaawIcacaGLPaaaaaGaeyypa0ZaaSaaaeaa cqaH4oqCdaWgaaqcfasaaiaaigdaaKqbagqaamaabmaabaGaeqiUde 3aaSbaaKqbGeaacaaIYaaajuaGbeaacqGHRaWkcaaIYaaacaGLOaGa ayzkaaaabaGaeqiUde3aaSbaaKqbGeaacaaIYaaajuaGbeaadaqada qaaiabeI7aXnaaBaaajuaibaGaaGymaaqcfayabaGaey4kaSIaaGOm aaGaayjkaiaawMcaaaaadaqadaqaamaalaaabaGaaGymaiabgUcaRi abeI7aXnaaBaaajuaibaGaaGymaaqcfayabaGaey4kaSIaeqiUde3a aSbaaKqbGeaacaaIXaaajuaGbeaacaaMc8UaamiEaaqaaiaaigdacq GHRaWkcqaH4oqCdaWgaaqcfasaaiaaikdaaKqbagqaaiabgUcaRiab eI7aXnaaBaaajuaibaGaaGOmaaqcfayabaGaaGPaVlaadIhaaaaaca GLOaGaayzkaaGaamyzamaaCaaabeqaaiabgkHiTmaabmaabaGaeqiU de3aaSbaaKqbGeaacaaIXaaajuaGbeaacqGHsislcqaH4oqCdaWgaa qcfasaaiaaikdaaKqbagqaaaGaayjkaiaawMcaaiaadIhaaaaaaa@7A74@   ; x>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaai4oai aabccacaWG4bGaeyOpa4JaaGimaaaa@3AA5@

Now

log f X ( x ) f Y ( x ) =log[ θ 1 ( θ 2 +2 ) θ 2 ( θ 1 +2 ) ]+log( 1+ θ 1 + θ 1 x 1+ θ 2 + θ 2 x )( θ 1 θ 2 )x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaciiBai aac+gacaGGNbWaaSqaaeaacaWGMbWaaSbaaKqbGeaacaWGybaajuaG beaadaqadaqaaiaadIhaaiaawIcacaGLPaaaaeaacaWGMbWaaSbaaK qbGeaacaWGzbaajuaGbeaadaqadaqaaiaadIhaaiaawIcacaGLPaaa aaGaeyypa0JaciiBaiaac+gacaGGNbWaamWaaeaadaWcaaqaaiabeI 7aXnaaBaaajuaibaGaaGymaaqcfayabaWaaeWaaeaacqaH4oqCdaWg aaqcfasaaiaaikdaaKqbagqaaiabgUcaRiaaikdaaiaawIcacaGLPa aaaeaacqaH4oqCdaWgaaqcfasaaiaaikdaaKqbagqaamaabmaabaGa eqiUde3aaSbaaKqbGeaacaaIXaaajuaGbeaacqGHRaWkcaaIYaaaca GLOaGaayzkaaaaaaGaay5waiaaw2faaiabgUcaRiGacYgacaGGVbGa ai4zamaabmaabaWaaSaaaeaacaaIXaGaey4kaSIaeqiUde3aaSbaaK qbGeaacaaIXaaajuaGbeaacqGHRaWkcqaH4oqCdaWgaaqcfasaaiaa igdaaKqbagqaaiaaykW7caWG4baabaGaaGymaiabgUcaRiabeI7aXn aaBaaajuaibaGaaGOmaaqcfayabaGaey4kaSIaeqiUde3aaSbaaKqb GeaacaaIYaaajuaGbeaacaaMc8UaamiEaaaaaiaawIcacaGLPaaacq GHsisldaqadaqaaiabeI7aXnaaBaaajuaibaGaaGymaaqcfayabaGa eyOeI0IaeqiUde3aaSbaaKqbGeaacaaIYaaajuaGbeaaaiaawIcaca GLPaaacaWG4baaaa@84AC@

This gives        d dx log f X ( x ) f Y ( x ) = θ 1 θ 2 ( 1+ θ 1 + θ 1 x )( 1+ θ 2 + θ 2 x ) ( θ 1 θ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGKbaabaGaamizaiaadIhaaaGaciiBaiaac+gacaGGNbWaaSqa aeaacaWGMbWaaSbaaKqbGeaacaWGybaajuaGbeaadaqadaqaaiaadI haaiaawIcacaGLPaaaaeaacaWGMbWaaSbaaKqbGeaacaWGzbaajuaG beaadaqadaqaaiaadIhaaiaawIcacaGLPaaaaaGaeyypa0ZaaSaaae aacqaH4oqCdaWgaaqcfasaaiaaigdaaKqbagqaaiabgkHiTiabeI7a XnaaBaaajuaibaGaaGOmaaqcfayabaaabaWaaeWaaeaacaaIXaGaey 4kaSIaeqiUde3aaSbaaKqbGeaacaaIXaaajuaGbeaacqGHRaWkcqaH 4oqCdaWgaaqcfasaaiaaigdaaKqbagqaaiaaykW7caWG4baacaGLOa GaayzkaaWaaeWaaeaacaaIXaGaey4kaSIaeqiUde3aaSbaaKqbGeaa caaIYaaajuaGbeaacqGHRaWkcqaH4oqCdaWgaaqcfasaaiaaikdaaK qbagqaaiaaykW7caWG4baacaGLOaGaayzkaaaaaiabgkHiTmaabmaa baGaeqiUde3aaSbaaKqbGeaacaaIXaaajuaGbeaacqGHsislcqaH4o qCdaWgaaqcfasaaiaaikdaaKqbagqaaaGaayjkaiaawMcaaaaa@7395@

 Thus for θ 1 θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde 3aaSbaaKqbGeaacaaIXaaajuaGbeaacqGHLjYScqaH4oqCdaWgaaqc fasaaiaaikdaaKqbagqaaaaa@3EE7@ , d dx log f X ( x ) f Y ( x ) <0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGKbaabaGaamizaiaadIhaaaGaciiBaiaac+gacaGGNbWaaSqa aeaacaWGMbWaaSbaaKqbGeaacaWGybaajuaGbeaadaqadaqaaiaadI haaiaawIcacaGLPaaaaeaacaWGMbWaaSbaaKqbGeaacaWGzbaajuaG beaadaqadaqaaiaadIhaaiaawIcacaGLPaaaaaGaeyipaWJaaGimaa aa@4859@ . This means that X lr Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaajuaibaGaamiBaiaadkhaaKqbagqaaiaadMfaaaa@3CB9@ and hence X hr Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaajuaibaGaamiAaiaadkhaaKqbagqaaiaadMfaaaa@3CB5@ , X mrl Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaajuaibaGaamyBaiaadkhacaWGSbaajuaGbeaacaWG zbaaaa@3DAB@ and X st Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaajuaibaGaam4CaiaadshaaKqbagqaaiaadMfaaaa@3CC2@ .

Mean Deviations

The amount of scatter in a population is measured to some extent by the totality of deviations usually from mean and median. These are known as the mean deviation about the mean and the mean deviation about the median defined by

δ 1 ( X )= 0 | xμ | f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaSbaaKqbGeaacaaIXaaajuaGbeaadaqadaqaaiaadIfaaiaawIca caGLPaaacqGH9aqpdaWdXbqaamaaemaabaGaamiEaiabgkHiTiabeY 7aTbGaay5bSlaawIa7aaqcfasaaiaaicdaaeaacqGHEisPaKqbakab gUIiYdGaaGPaVlaadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaaca WGKbGaamiEaaaa@5009@ and δ 2 ( X )= 0 | xM | f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaSbaaKqbGeaacaaIYaaajuaGbeaadaqadaqaaiaadIfaaiaawIca caGLPaaacqGH9aqpdaWdXbqaamaaemaabaGaamiEaiabgkHiTiaad2 eaaiaawEa7caGLiWoaaKqbGeaacaaIWaaabaGaeyOhIukajuaGcqGH RiI8aiaaykW7caWGMbWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaam izaiaadIhaaaa@4F26@ , respectively, where μ=E( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 Maeyypa0JaamyramaabmaabaGaamiwaaGaayjkaiaawMcaaaaa@3C70@  and M=Median ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamytai abg2da9iaab2eacaqGLbGaaeizaiaabMgacaqGHbGaaeOBaiaabcca daqadaqaaiaadIfaaiaawIcacaGLPaaaaaa@40C5@ . The measures δ 1 ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaSbaaKqbGeaacaaIXaaajuaGbeaadaqadaqaaiaadIfaaiaawIca caGLPaaaaaa@3C27@  and δ 2 ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaSbaaKqbGeaacaaIYaaajuaGbeaadaqadaqaaiaadIfaaiaawIca caGLPaaaaaa@3C28@ can be calculated using the relationships

δ 1 ( X )= 0 μ ( μx ) f( x )dx+ μ ( xμ ) f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaSbaaKqbGeaacaaIXaaajuaGbeaadaqadaqaaiaadIfaaiaawIca caGLPaaacqGH9aqpdaWdXbqaamaabmaabaGaeqiVd0MaeyOeI0Iaam iEaaGaayjkaiaawMcaaaqcfasaaiaaicdaaeaacqaH8oqBaKqbakab gUIiYdGaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaiaadsgaca WG4bGaey4kaSYaa8qCaeaadaqadaqaaiaadIhacqGHsislcqaH8oqB aiaawIcacaGLPaaaaKqbGeaacqaH8oqBaeaacqGHEisPaKqbakabgU IiYdGaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaiaadsgacaWG 4baaaa@5EC0@

=μF( μ ) 0 μ xf( x )dx μ[ 1F( μ ) ]+ μ x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 JaeqiVd0MaamOramaabmaabaGaeqiVd0gacaGLOaGaayzkaaGaeyOe I0Yaa8qCaeaacaWG4bGaaGPaVlaadAgadaqadaqaaiaadIhaaiaawI cacaGLPaaacaWGKbGaamiEaaqcfasaaiaaicdaaeaacqaH8oqBaKqb akabgUIiYdGaeyOeI0IaeqiVd02aamWaaeaacaaIXaGaeyOeI0Iaam OramaabmaabaGaeqiVd0gacaGLOaGaayzkaaaacaGLBbGaayzxaaGa ey4kaSYaa8qCaeaacaWG4bGaaGPaVdqcfasaaiabeY7aTbqaaiabg6 HiLcqcfaOaey4kIipacaWGMbWaaeWaaeaacaWG4baacaGLOaGaayzk aaGaamizaiaadIhaaaa@64CF@

=2μF( μ )2μ+2 μ x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 JaaGOmaiabeY7aTjaadAeadaqadaqaaiabeY7aTbGaayjkaiaawMca aiabgkHiTiaaikdacqaH8oqBcqGHRaWkcaaIYaWaa8qCaeaacaWG4b GaaGPaVdqcfasaaiabeY7aTbqaaiabg6HiLcqcfaOaey4kIipacaWG MbWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaamizaiaadIhaaaa@5116@

=2μF( μ )2 0 μ x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 JaaGOmaiabeY7aTjaadAeadaqadaqaaiabeY7aTbGaayjkaiaawMca aiabgkHiTiaaikdadaWdXbqaaiaadIhacaaMc8oajuaibaGaaGimaa qaaiabeY7aTbqcfaOaey4kIipacaWGMbWaaeWaaeaacaWG4baacaGL OaGaayzkaaGaamizaiaadIhaaaa@4D0B@                                                            (7.1)

and

δ 2 ( X )= 0 M ( Mx ) f( x )dx+ M ( xM ) f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaSbaaKqbGeaacaaIYaaajuaGbeaadaqadaqaaiaadIfaaiaawIca caGLPaaacqGH9aqpdaWdXbqaamaabmaabaGaamytaiabgkHiTiaadI haaiaawIcacaGLPaaaaKqbGeaacaaIWaaabaGaamytaaqcfaOaey4k IipacaWGMbWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaamizaiaadI hacqGHRaWkdaWdXbqaamaabmaabaGaamiEaiabgkHiTiaad2eaaiaa wIcacaGLPaaaaKqbGeaacaWGnbaabaGaeyOhIukajuaGcqGHRiI8ai aadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaacaWGKbGaamiEaaaa @5B31@

=MF( M ) 0 M xf( x )dx M[ 1F( M ) ]+ M x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 JaamytaiaaykW7caWGgbWaaeWaaeaacaWGnbaacaGLOaGaayzkaaGa eyOeI0Yaa8qCaeaacaWG4bGaaGPaVlaadAgadaqadaqaaiaadIhaai aawIcacaGLPaaacaWGKbGaamiEaaqcfasaaiaaicdaaeaacaWGnbaa juaGcqGHRiI8aiabgkHiTiaad2eadaWadaqaaiaaigdacqGHsislca WGgbWaaeWaaeaacaWGnbaacaGLOaGaayzkaaaacaGLBbGaayzxaaGa ey4kaSYaa8qCaeaacaWG4bGaaGPaVdqcfasaaiaad2eaaeaacqGHEi sPaKqbakabgUIiYdGaamOzamaabmaabaGaamiEaaGaayjkaiaawMca aiaadsgacaWG4baaaa@6102@

=μ+2 M x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 JaeyOeI0IaeqiVd0Maey4kaSIaaGOmamaapehabaGaamiEaiaaykW7 aKqbGeaacaWGnbaabaGaeyOhIukajuaGcqGHRiI8aiaadAgadaqada qaaiaadIhaaiaawIcacaGLPaaacaWGKbGaamiEaaaa@48FA@

=μ2 0 M x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 JaeqiVd0MaeyOeI0IaaGOmamaapehabaGaamiEaiaaykW7aKqbGeaa caaIWaaabaGaamytaaqcfaOaey4kIipacaWGMbWaaeWaaeaacaWG4b aacaGLOaGaayzkaaGaamizaiaadIhaaaa@4761@                                                                    (7.2)

Using p.d.f. (2.1) and expression for the mean of Garima distribution, we get

0 μ x f 7 ( x;θ )dx=μ { θ 2 μ 2 +( θ 2 +3θ )μ+( θ+3 ) } e θμ θ( θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa8qCae aacaWG4bGaaGPaVdqcfasaaiaaicdaaeaacqaH8oqBaKqbakabgUIi YdGaamOzamaaBaaajuaibaGaaG4naaqcfayabaWaaeWaaeaacaWG4b Gaai4oaiabeI7aXbGaayjkaiaawMcaaiaadsgacaWG4bGaeyypa0Ja eqiVd0MaeyOeI0YaaSaaaeaadaGadaqaaiabeI7aXnaaCaaabeqcfa saaiaaikdaaaqcfaOaeqiVd02aaWbaaeqajuaibaGaaGOmaaaajuaG cqGHRaWkdaqadaqaaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfa Oaey4kaSIaaG4maiabeI7aXbGaayjkaiaawMcaaiabeY7aTjabgUca RmaabmaabaGaeqiUdeNaey4kaSIaaG4maaGaayjkaiaawMcaaaGaay 5Eaiaaw2haaiaadwgadaahaaqabKqbGeaacqGHsislcqaH4oqCcaaM c8UaeqiVd0gaaaqcfayaaiabeI7aXnaabmaabaGaeqiUdeNaey4kaS IaaGOmaaGaayjkaiaawMcaaaaaaaa@7342@                            (7.3)

0 M x f 7 ( x;θ )dx=μ { θ 2 M 2 +( θ 2 +3θ )M+( θ+3 ) } e θM θ( θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa8qCae aacaWG4bGaaGPaVdqcfasaaiaaicdaaeaacaWGnbaajuaGcqGHRiI8 aiaadAgadaWgaaqcfasaaiaaiEdaaKqbagqaamaabmaabaGaamiEai aacUdacqaH4oqCaiaawIcacaGLPaaacaWGKbGaamiEaiabg2da9iab eY7aTjabgkHiTmaalaaabaWaaiWaaeaacqaH4oqCdaahaaqabKqbGe aacaaIYaaaaKqbakaad2eadaahaaqabKqbGeaacaaIYaaaaKqbakab gUcaRmaabmaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcq GHRaWkcaaIZaGaeqiUdehacaGLOaGaayzkaaGaamytaiabgUcaRmaa bmaabaGaeqiUdeNaey4kaSIaaG4maaGaayjkaiaawMcaaaGaay5Eai aaw2haaiaadwgadaahaaqabKqbGeaacqGHsislcqaH4oqCcaaMc8Ua amytaaaaaKqbagaacqaH4oqCdaqadaqaaiabeI7aXjabgUcaRiaaik daaiaawIcacaGLPaaaaaaaaa@6FB2@                            (7.4)

Using expressions from (7.1), (7.2), (7.3), and (7.4), the mean deviation about mean, δ 1 ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaSbaaKqbGeaacaaIXaaajuaGbeaadaqadaqaaiaadIfaaiaawIca caGLPaaaaaa@3C27@  and the mean deviation about median, δ 2 ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaSbaaKqbGeaacaaIYaaajuaGbeaadaqadaqaaiaadIfaaiaawIca caGLPaaaaaa@3C28@  of Garima distribution are obtained as

δ 1 ( X )= ( 2θμ+θ+3 ) e θμ θ( θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaSbaaKqbGeaacaaIXaaajuaGbeaadaqadaqaaiaadIfaaiaawIca caGLPaaacqGH9aqpdaWcaaqaamaabmaabaGaaGOmaiabeI7aXjaayk W7cqaH8oqBcqGHRaWkcqaH4oqCcqGHRaWkcaaIZaaacaGLOaGaayzk aaGaamyzamaaCaaabeqcfasaaiabgkHiTiabeI7aXjaaykW7cqaH8o qBaaaajuaGbaGaeqiUde3aaeWaaeaacqaH4oqCcqGHRaWkcaaIYaaa caGLOaGaayzkaaaaaaaa@56EF@                                                                        (7.5)

δ 2 ( X )= 2{ θ 2 M 2 +( θ 2 +3θ )M+( θ+3 ) } e θM θ( θ+2 ) μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaSbaaKqbGeaacaaIYaaajuaGbeaadaqadaqaaiaadIfaaiaawIca caGLPaaacqGH9aqpdaWcaaqaaiaaikdadaGadaqaaiabeI7aXnaaCa aabeqcfasaaiaaikdaaaqcfaOaaGPaVlaad2eadaahaaqabKqbGeaa caaIYaaaaKqbakabgUcaRmaabmaabaGaeqiUde3aaWbaaeqajuaiba GaaGOmaaaajuaGcqGHRaWkcaaIZaGaeqiUdehacaGLOaGaayzkaaGa amytaiabgUcaRmaabmaabaGaeqiUdeNaey4kaSIaaG4maaGaayjkai aawMcaaaGaay5Eaiaaw2haaiaadwgadaahaaqabKqbGeaacqGHsisl cqaH4oqCcaaMc8UaamytaaaaaKqbagaacqaH4oqCdaqadaqaaiabeI 7aXjabgUcaRiaaikdaaiaawIcacaGLPaaaaaGaeyOeI0IaeqiVd0ga aa@6712@                                   (7.6)

Order Statistics

Let X 1 , X 2 ,..., X n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwam aaBaaajuaibaGaaGymaaqcfayabaGaaiilaiaaykW7caWGybWaaSba aKqbGeaacaaIYaaajuaGbeaacaGGSaGaaGPaVlaac6cacaGGUaGaai OlaiaacYcacaaMc8UaamiwamaaBaaajuaibaGaamOBaaqcfayabaaa aa@46E3@  be a random sample of size n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOBaa aa@3777@  from Garima distribution (2.1). Let X ( 1 ) < X ( 2 ) <...< X ( n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwam aaBaaajuaibaqcfa4aaeWaaKqbGeaacaaIXaaacaGLOaGaayzkaaaa beaajuaGcqGH8aapcaWGybWaaSbaaKqbGeaajuaGdaqadaqcfasaai aaikdaaiaawIcacaGLPaaaaeqaaKqbakabgYda8iaaykW7caaMc8Ua aiOlaiaac6cacaGGUaGaaGPaVlaaykW7cqGH8aapcaWGybWaaSbaaK qbGeaajuaGdaqadaqcfasaaiaad6gaaiaawIcacaGLPaaaaKqbagqa aaaa@5039@ denote the corresponding order statistics. The p.d.f. and the c.d.f. of the k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Aaa aa@3774@ th order statistic, say Y= X ( k ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamywai abg2da9iaadIfadaWgaaqcfasaaKqbaoaabmaajuaibaGaam4AaaGa ayjkaiaawMcaaaqcfayabaaaaa@3D57@ are given by

f Y ( y )= n! ( k1 )!( nk )! F k1 ( y ) { 1F( y ) } nk f( y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaamywaaqcfayabaWaaeWaaeaacaWG5baacaGLOaGa ayzkaaGaeyypa0ZaaSaaaeaacaWGUbGaaiyiaaqaamaabmaabaGaam 4AaiabgkHiTiaaigdaaiaawIcacaGLPaaacaGGHaGaaGPaVpaabmaa baGaamOBaiabgkHiTiaadUgaaiaawIcacaGLPaaacaGGHaaaaiaayk W7caWGgbWaaWbaaeqajuaibaGaam4AaiabgkHiTiaaigdaaaqcfa4a aeWaaeaacaWG5baacaGLOaGaayzkaaWaaiWaaeaacaaIXaGaeyOeI0 IaamOramaabmaabaGaamyEaaGaayjkaiaawMcaaaGaay5Eaiaaw2ha amaaCaaabeqcfasaaiaad6gacqGHsislcaWGRbaaaKqbakaadAgada qadaqaaiaadMhaaiaawIcacaGLPaaaaaa@604C@

= n! ( k1 )!( nk )! l=0 nk ( nk l ) ( 1 ) l F k+l1 ( y )f( y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacaWGUbGaaiyiaaqaamaabmaabaGaam4AaiabgkHiTiaa igdaaiaawIcacaGLPaaacaGGHaGaaGPaVpaabmaabaGaamOBaiabgk HiTiaadUgaaiaawIcacaGLPaaacaGGHaaaaiaaykW7daaeWbqaamaa bmaabaqbaeqabiqaaaqaaiaad6gacqGHsislcaWGRbaabaGaamiBaa aaaiaawIcacaGLPaaaaKqbGeaacaWGSbGaeyypa0JaaGimaaqaaiaa d6gacqGHsislcaWGRbaajuaGcqGHris5amaabmaabaGaeyOeI0IaaG ymaaGaayjkaiaawMcaamaaCaaabeqcfasaaiaadYgaaaqcfaOaamOr amaaCaaabeqcfasaaiaadUgacqGHRaWkcaWGSbGaeyOeI0IaaGymaa aajuaGdaqadaqaaiaadMhaaiaawIcacaGLPaaacaWGMbWaaeWaaeaa caWG5baacaGLOaGaayzkaaaaaa@64D7@

and

F Y ( y )= j=k n ( n j ) F j ( y ) { 1F( y ) } nj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaamywaaqcfayabaWaaeWaaeaacaWG5baacaGLOaGa ayzkaaGaeyypa0ZaaabCaeaadaqadaqaauaabeqaceaaaeaacaWGUb aabaGaamOAaaaaaiaawIcacaGLPaaaaKqbGeaacaWGQbGaeyypa0Ja am4Aaaqaaiaad6gaaKqbakabggHiLdGaaGPaVlaadAeadaahaaqabK qbGeaacaWGQbaaaKqbaoaabmaabaGaamyEaaGaayjkaiaawMcaamaa cmaabaGaaGymaiabgkHiTiaadAeadaqadaqaaiaadMhaaiaawIcaca GLPaaaaiaawUhacaGL9baadaahaaqabKqbGeaacaWGUbGaeyOeI0Ia amOAaaaaaaa@57C1@

= j=k n l=0 nj ( n j ) ( nj l ) ( 1 ) l F j+l ( y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaabCaeaadaaeWbqaamaabmaabaqbaeqabiqaaaqaaiaad6gaaeaa caWGQbaaaaGaayjkaiaawMcaaaqcfasaaiaadYgacqGH9aqpcaaIWa aabaGaamOBaiabgkHiTiaadQgaaKqbakabggHiLdWaaeWaaeaafaqa beGabaaabaGaamOBaiabgkHiTiaadQgaaeaacaWGSbaaaaGaayjkai aawMcaaaqcfasaaiaadQgacqGH9aqpcaWGRbaabaGaamOBaaqcfaOa eyyeIuoacaaMc8+aaeWaaeaacqGHsislcaaIXaaacaGLOaGaayzkaa WaaWbaaeqajuaibaGaamiBaaaajuaGcaWGgbWaaWbaaeqajuaibaGa amOAaiabgUcaRiaadYgaaaqcfa4aaeWaaeaacaWG5baacaGLOaGaay zkaaaaaa@5CF9@ ,

respectively, for k=1,2,3,...,n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Aai abg2da9iaaigdacaGGSaGaaGOmaiaacYcacaaIZaGaaiilaiaac6ca caGGUaGaaiOlaiaacYcacaWGUbaaaa@4077@ .

 Thus,  the p.d.f. and the c.d.f of k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@36E6@ th  order statistics of Garima distribution are given by

f Y ( y )= n!θ( 1+θ+θx ) e θx ( θ+2 )( k1 )!( nk )! l=0 nk ( nk l ) × [ 1 θx+( θ+2 ) θ+2 e θx ] k+l1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaamywaaqcfayabaWaaeWaaeaacaWG5baacaGLOaGa ayzkaaGaeyypa0ZaaSaaaeaacaWGUbGaaiyiaiabeI7aXnaabmaaba GaaGymaiabgUcaRiabeI7aXjabgUcaRiabeI7aXjaaykW7caWG4baa caGLOaGaayzkaaGaamyzamaaCaaabeqcfasaaiabgkHiTiabeI7aXj aadIhaaaaajuaGbaWaaeWaaeaacqaH4oqCcqGHRaWkcaaIYaaacaGL OaGaayzkaaWaaeWaaeaacaWGRbGaeyOeI0IaaGymaaGaayjkaiaawM caaiaacgcacaaMc8+aaeWaaeaacaWGUbGaeyOeI0Iaam4AaaGaayjk aiaawMcaaiaacgcaaaGaaGPaVpaaqahabaWaaeWaaeaafaqabeGaba aabaGaamOBaiabgkHiTiaadUgaaeaacaWGSbaaaaGaayjkaiaawMca aaqcfasaaiaadYgacqGH9aqpcaaIWaaabaGaamOBaiabgkHiTiaadU gaaKqbakabggHiLdGaey41aq7aamWaaeaacaaIXaGaeyOeI0YaaSaa aeaacqaH4oqCcaWG4bGaey4kaSYaaeWaaeaacqaH4oqCcqGHRaWkca aIYaaacaGLOaGaayzkaaaabaGaeqiUdeNaey4kaSIaaGOmaaaacaWG LbWaaWbaaeqajuaibaGaeyOeI0IaeqiUdeNaamiEaaaaaKqbakaawU facaGLDbaadaahaaqabKqbGeaacaWGRbGaey4kaSIaamiBaiabgkHi Tiaaigdaaaaaaa@8A98@

and

F Y ( y )= j=k n l=0 nj ( n j ) ( nj l ) ( 1 ) l [ 1 θx+( θ+2 ) θ+2 e θx ] j+l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaamywaaqcfayabaWaaeWaaeaacaWG5baacaGLOaGa ayzkaaGaeyypa0ZaaabCaeaadaaeWbqaamaabmaabaqbaeqabiqaaa qaaiaad6gaaeaacaWGQbaaaaGaayjkaiaawMcaaaqcfasaaiaadYga cqGH9aqpcaaIWaaabaGaamOBaiabgkHiTiaadQgaaKqbakabggHiLd WaaeWaaeaafaqabeGabaaabaGaamOBaiabgkHiTiaadQgaaeaacaWG SbaaaaGaayjkaiaawMcaaaqcfasaaiaadQgacqGH9aqpcaWGRbaaba GaamOBaaqcfaOaeyyeIuoacaaMc8+aaeWaaeaacqGHsislcaaIXaaa caGLOaGaayzkaaWaaWbaaeqajuaibaGaamiBaaaajuaGdaWadaqaai aaigdacqGHsisldaWcaaqaaiabeI7aXjaadIhacqGHRaWkdaqadaqa aiabeI7aXjabgUcaRiaaikdaaiaawIcacaGLPaaaaeaacqaH4oqCcq GHRaWkcaaIYaaaaiaadwgadaahaaqabKqbGeaacqGHsislcqaH4oqC caWG4baaaaqcfaOaay5waiaaw2faamaaCaaabeqcfasaaiaadQgacq GHRaWkcaWGSbaaaaaa@72FE@

Bonferroni and Lorenz Curves

The Bonferroni and Lorenz curves [7] and Bonferroni and Gini indices have applications not only in economics to study income and poverty, but also in other fields like reliability, demography, insurance and medicine. The Bonferroni and Lorenz curves are defined as

B( p )= 1 pμ 0 q xf( x ) dx= 1 pμ [ 0 xf( x )dx q xf( x ) dx ]= 1 pμ [ μ q xf( x ) dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOqam aabmaabaGaamiCaaGaayjkaiaawMcaaiabg2da9maalaaabaGaaGym aaqaaiaadchacqaH8oqBaaWaa8qCaeaacaWG4bGaaGPaVlaadAgada qadaqaaiaadIhaaiaawIcacaGLPaaacaaMc8oajuaibaGaaGimaaqa aiaadghaaKqbakabgUIiYdGaamizaiaadIhacqGH9aqpdaWcaaqaai aaigdaaeaacaWGWbGaeqiVd0gaamaadmaabaWaa8qCaeaacaWG4bGa aGPaVlaadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaacaWGKbGaam iEaiabgkHiTaqcfasaaiaaicdaaeaacqGHEisPaKqbakabgUIiYdWa a8qCaeaacaWG4bGaaGPaVlaadAgadaqadaqaaiaadIhaaiaawIcaca GLPaaaaKqbGeaacaWGXbaabaGaeyOhIukajuaGcqGHRiI8aiaaykW7 caWGKbGaamiEaaGaay5waiaaw2faaiabg2da9maalaaabaGaaGymaa qaaiaadchacqaH8oqBaaWaamWaaeaacqaH8oqBcqGHsisldaWdXbqa aiaadIhacaaMc8UaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaa qcfasaaiaadghaaeaacqGHEisPaKqbakabgUIiYdGaaGPaVlaadsga caWG4baacaGLBbGaayzxaaaaaa@87AB@        (9.1)

and L( p )= 1 μ 0 q xf( x ) dx= 1 μ [ 0 xf( x )dx q xf( x ) dx ]= 1 μ [ μ q xf( x ) dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitam aabmaabaGaamiCaaGaayjkaiaawMcaaiabg2da9maalaaabaGaaGym aaqaaiabeY7aTbaadaWdXbqaaiaadIhacaaMc8UaamOzamaabmaaba GaamiEaaGaayjkaiaawMcaaaqcfasaaiaaicdaaeaacaWGXbaajuaG cqGHRiI8aiaaykW7caWGKbGaamiEaiabg2da9maalaaabaGaaGymaa qaaiabeY7aTbaadaWadaqaamaapehabaGaamiEaiaaykW7caWGMbWa aeWaaeaacaWG4baacaGLOaGaayzkaaGaamizaiaadIhacqGHsislaK qbGeaacaaIWaaabaGaeyOhIukajuaGcqGHRiI8amaapehabaGaamiE aiaaykW7caWGMbWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaaGPaVd qcfasaaiaadghaaeaacqGHEisPaKqbakabgUIiYdGaamizaiaadIha aiaawUfacaGLDbaacqGH9aqpdaWcaaqaaiaaigdaaeaacqaH8oqBaa WaamWaaeaacqaH8oqBcqGHsisldaWdXbqaaiaadIhacaaMc8UaamOz amaabmaabaGaamiEaaGaayjkaiaawMcaaaqcfasaaiaadghaaeaacq GHEisPaKqbakabgUIiYdGaaGPaVlaadsgacaWG4baacaGLBbGaayzx aaaaaa@84D6@          (9.2)

respectively or equivalently

B( p )= 1 pμ 0 p F 1 ( x ) dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOqam aabmaabaGaamiCaaGaayjkaiaawMcaaiabg2da9maalaaabaGaaGym aaqaaiaadchacqaH8oqBaaWaa8qCaeaacaWGgbWaaWbaaeqajuaiba GaeyOeI0IaaGymaaaajuaGdaqadaqaaiaadIhaaiaawIcacaGLPaaa aKqbGeaacaaIWaaabaGaamiCaaqcfaOaey4kIipacaaMc8Uaamizai aadIhaaaa@4C49@                                                                      (9.3)

and    L( p )= 1 μ 0 p F 1 ( x ) dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitam aabmaabaGaamiCaaGaayjkaiaawMcaaiabg2da9maalaaabaGaaGym aaqaaiabeY7aTbaadaWdXbqaaiaadAeadaahaaqabKqbGeaacqGHsi slcaaIXaaaaKqbaoaabmaabaGaamiEaaGaayjkaiaawMcaaaqcfasa aiaaicdaaeaacaWGWbaajuaGcqGHRiI8aiaaykW7caWGKbGaamiEaa aa@4B5E@                                                                       (9.4)

respectively, where μ=E( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 Maeyypa0JaamyramaabmaabaGaamiwaaGaayjkaiaawMcaaaaa@3C70@ and q= F 1 ( p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyCai abg2da9iaadAeadaahaaqabKqbGeaacqGHsislcaaIXaaaaKqbaoaa bmaabaGaamiCaaGaayjkaiaawMcaaaaa@3E4F@ .

The Bonferroni and Gini indices are thus defined as

B=1 0 1 B( p ) dp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOqai abg2da9iaaigdacqGHsisldaWdXbqaaiaadkeadaqadaqaaiaadcha aiaawIcacaGLPaaaaKqbGeaacaaIWaaabaGaaGymaaqcfaOaey4kIi pacaaMc8Uaamizaiaadchaaaa@4529@                                                                        (9.5)

and G=12 0 1 L( p ) dp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4rai abg2da9iaaigdacqGHsislcaaIYaWaa8qCaeaacaWGmbWaaeWaaeaa caWGWbaacaGLOaGaayzkaaGaaGPaVdqcfasaaiaaicdaaeaacaaIXa aajuaGcqGHRiI8aiaadsgacaWGWbaaaa@45F4@                                                                (9.6)

respectively.

Using p.d.f. (2.1), we get

q x f 7 ( x;θ ) dx= { θ 2 q 2 +( θ 2 +3θ )q+( θ+3 ) } e θq θ( θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa8qCae aacaWG4bGaaGPaVlaadAgadaWgaaqcfasaaiaaiEdaaKqbagqaamaa bmaabaGaamiEaiaacUdacqaH4oqCaiaawIcacaGLPaaaaKqbGeaaca WGXbaabaGaeyOhIukajuaGcqGHRiI8aiaaykW7caWGKbGaamiEaiab g2da9maalaaabaWaaiWaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYa aaaKqbakaaykW7caWGXbWaaWbaaeqajuaibaGaaGOmaaaajuaGcqGH RaWkdaqadaqaaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey 4kaSIaaG4maiabeI7aXbGaayjkaiaawMcaaiaadghacqGHRaWkdaqa daqaaiabeI7aXjabgUcaRiaaiodaaiaawIcacaGLPaaaaiaawUhaca GL9baacaWGLbWaaWbaaeqajuaibaGaeyOeI0IaeqiUdeNaamyCaaaa aKqbagaacqaH4oqCdaqadaqaaiabeI7aXjabgUcaRiaaikdaaiaawI cacaGLPaaaaaaaaa@6FE1@                                                    (9.7)

Now using equation (8.7) in (8.1) and (8.2), we get

B( p )= 1 p [ 1 { θ 2 q 2 +( θ 2 +3θ )q+( θ+3 ) } e θq θ+3 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOqam aabmaabaGaamiCaaGaayjkaiaawMcaaiabg2da9maalaaabaGaaGym aaqaaiaadchaaaWaamWaaeaacaaIXaGaeyOeI0YaaSaaaeaadaGada qaaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaaGPaVlaadgha daahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRmaabmaabaGaeqiUde 3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaIZaGaeqiUdeha caGLOaGaayzkaaGaamyCaiabgUcaRmaabmaabaGaeqiUdeNaey4kaS IaaG4maaGaayjkaiaawMcaaaGaay5Eaiaaw2haaiaadwgadaahaaqa bKqbGeaacqGHsislcqaH4oqCcaWGXbaaaaqcfayaaiabeI7aXjabgU caRiaaiodaaaaacaGLBbGaayzxaaaaaa@6251@                                                (9.8)

and  L( p )=1 { θ 2 q 2 +( θ 2 +3θ )q+( θ+3 ) } e θq θ+3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitam aabmaabaGaamiCaaGaayjkaiaawMcaaiabg2da9iaaigdacqGHsisl daWcaaqaamaacmaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaaju aGcaaMc8UaamyCamaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSYa aeWaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRi aaiodacqaH4oqCaiaawIcacaGLPaaacaWGXbGaey4kaSYaaeWaaeaa cqaH4oqCcqGHRaWkcaaIZaaacaGLOaGaayzkaaaacaGL7bGaayzFaa GaamyzamaaCaaabeqcfasaaiabgkHiTiabeI7aXjaadghaaaaajuaG baGaeqiUdeNaey4kaSIaaG4maaaaaaa@5EA9@                                                        (9.9)

Now using equations (9.8) and (9.9) in (9.5) and (9.6), the Bonferroni and Gini indices of Garima distribution (2.1) are obtained as

B=1 { θ 2 q 2 +( θ 2 +3θ )q+( θ+3 ) } e θq θ+3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOqai abg2da9iaaigdacqGHsisldaWcaaqaamaacmaabaGaeqiUde3aaWba aeqajuaibaGaaGOmaaaajuaGcaaMc8UaamyCamaaCaaabeqcfasaai aaikdaaaqcfaOaey4kaSYaaeWaaeaacqaH4oqCdaahaaqabKqbGeaa caaIYaaaaKqbakabgUcaRiaaiodacqaH4oqCaiaawIcacaGLPaaaca WGXbGaey4kaSYaaeWaaeaacqaH4oqCcqGHRaWkcaaIZaaacaGLOaGa ayzkaaaacaGL7bGaayzFaaGaamyzamaaCaaabeqcfasaaiabgkHiTi abeI7aXjaadghaaaaajuaGbaGaeqiUdeNaey4kaSIaaG4maaaaaaa@5C21@                                                       (9.10)

G=1+ 2{ θ 2 q 2 +( θ 2 +3θ )q+( θ+3 ) } e θq θ+3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4rai abg2da9iabgkHiTiaaigdacqGHRaWkdaWcaaqaaiaaikdadaGadaqa aiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaaGPaVlaadghada ahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRmaabmaabaGaeqiUde3a aWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaIZaGaeqiUdehaca GLOaGaayzkaaGaamyCaiabgUcaRmaabmaabaGaeqiUdeNaey4kaSIa aG4maaGaayjkaiaawMcaaaGaay5Eaiaaw2haaiaadwgadaahaaqabK qbGeaacqGHsislcqaH4oqCcaWGXbaaaaqcfayaaiabeI7aXjabgUca Riaaiodaaaaaaa@5DC4@                                                   (9.11)

Renyi Entropy

Entropy of a random variable X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwaa aa@3761@ is a measure of variation of uncertainty. A popular entropy measure is Renyi entropy [8]. If X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwaa aa@3761@ is a continuous random variable having probability density function f( . ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaaiOlaaGaayjkaiaawMcaaaaa@39AA@ , then Renyi entropy is defined as

T R ( γ )= 1 1γ log{ f γ ( x )dx } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaBaaajuaibaGaamOuaaqcfayabaWaaeWaaeaacqaHZoWzaiaawIca caGLPaaacqGH9aqpdaWcaaqaaiaaigdaaeaacaaIXaGaeyOeI0Iaeq 4SdCgaaiGacYgacaGGVbGaai4zamaacmaabaWaa8qaaeaacaWGMbWa aWbaaeqajuaibaGaeq4SdCgaaKqbaoaabmaabaGaamiEaaGaayjkai aawMcaaiaadsgacaWG4baabeqabiabgUIiYdaacaGL7bGaayzFaaaa aa@5021@

where γ>0andγ1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC MaeyOpa4JaaGimaiaaykW7caaMc8UaaGPaVlaabggacaqGUbGaaeiz aiaaykW7caaMc8UaaGPaVlabeo7aNjabgcMi5kaaigdaaaa@4A14@ .

Thus, the Renyi entropy for the Garima distribution (2.1) is obtained as

T R ( γ )= 1 1γ log[ 0 θ γ ( θ+2 ) γ ( 1+θ+θx ) γ e θγx dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivam aaBaaajuaibaGaamOuaaqcfayabaWaaeWaaeaacqaHZoWzaiaawIca caGLPaaacqGH9aqpdaWcaaqaaiaaigdaaeaacaaIXaGaeyOeI0Iaeq 4SdCgaaiGacYgacaGGVbGaai4zamaadmaabaWaa8qCaeaadaWcaaqa aiabeI7aXnaaCaaabeqcfasaaiabeo7aNbaaaKqbagaadaqadaqaai abeI7aXjabgUcaRiaaikdaaiaawIcacaGLPaaadaahaaqabKqbGeaa cqaHZoWzaaaaaKqbaoaabmaabaGaaGymaiabgUcaRiabeI7aXjabgU caRiabeI7aXjaaykW7caWG4baacaGLOaGaayzkaaWaaWbaaeqajuai baGaeq4SdCgaaKqbakaadwgadaahaaqabKqbGeaacqGHsislcqaH4o qCcaaMc8Uaeq4SdCMaaGPaVlaadIhaaaqcfaOaamizaiaadIhaaKqb GeaacaaIWaaabaGaeyOhIukajuaGcqGHRiI8aaGaay5waiaaw2faaa aa@6F94@

= 1 1γ log[ 0 θ γ ( 1+θ ) γ ( θ+2 ) γ ( 1+ θ θ+1 x ) γ e θγx dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacaaIXaaabaGaaGymaiabgkHiTiabeo7aNbaaciGGSbGa ai4BaiaacEgadaWadaqaamaapehabaWaaSaaaeaacqaH4oqCdaahaa qabKqbGeaacqaHZoWzaaqcfa4aaeWaaeaacaaIXaGaey4kaSIaeqiU dehacaGLOaGaayzkaaWaaWbaaeqajuaibaGaeq4SdCgaaaqcfayaam aabmaabaGaeqiUdeNaey4kaSIaaGOmaaGaayjkaiaawMcaamaaCaaa beqcfasaaiabeo7aNbaaaaqcfa4aaeWaaeaacaaIXaGaey4kaSYaaS aaaeaacqaH4oqCaeaacqaH4oqCcqGHRaWkcaaIXaaaaiaadIhaaiaa wIcacaGLPaaadaahaaqabKqbGeaacqaHZoWzaaqcfaOaamyzamaaCa aabeqcfasaaiabgkHiTiabeI7aXjaaykW7cqaHZoWzcaaMc8UaamiE aaaajuaGcaWGKbGaamiEaaqcfasaaiaaicdaaeaacqGHEisPaKqbak abgUIiYdaacaGLBbGaayzxaaaaaa@7078@

= 1 1γ log[ 0 θ γ ( 1+θ ) γ ( θ+2 ) γ j=0 ( γ j ) ( θ θ+1 x ) j e θγx dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacaaIXaaabaGaaGymaiabgkHiTiabeo7aNbaaciGGSbGa ai4BaiaacEgadaWadaqaamaapehabaWaaSaaaeaacqaH4oqCdaahaa qabKqbGeaacqaHZoWzaaqcfa4aaeWaaeaacaaIXaGaey4kaSIaeqiU dehacaGLOaGaayzkaaWaaWbaaeqajuaibaGaeq4SdCgaaaqcfayaam aabmaabaGaeqiUdeNaey4kaSIaaGOmaaGaayjkaiaawMcaamaaCaaa beqcfasaaiabeo7aNbaaaaqcfa4aaabCaeaadaqadaqaauaabeqace aaaeaacqaHZoWzaeaacaWGQbaaaaGaayjkaiaawMcaaaqcfasaaiaa dQgacqGH9aqpcaaIWaaabaGaeyOhIukajuaGcqGHris5aiaaykW7da qadaqaamaalaaabaGaeqiUdehabaGaeqiUdeNaey4kaSIaaGymaaaa caWG4baacaGLOaGaayzkaaWaaWbaaeqajuaibaGaamOAaaaajuaGca WGLbWaaWbaaeqajuaibaGaeyOeI0IaeqiUdeNaaGPaVlabeo7aNjaa ykW7caWG4baaaKqbakaadsgacaWG4baajuaibaGaaGimaaqaaiabg6 HiLcqcfaOaey4kIipaaiaawUfacaGLDbaaaaa@7AE0@

= 1 1γ log[ j=0 ( γ j ) θ γ+j ( 1+θ ) γj ( θ+2 ) γ 0 e θγx x j+11 dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacaaIXaaabaGaaGymaiabgkHiTiabeo7aNbaaciGGSbGa ai4BaiaacEgadaWadaqaamaaqahabaWaaeWaaeaafaqabeGabaaaba Gaeq4SdCgabaGaamOAaaaaaiaawIcacaGLPaaaaKqbGeaacaWGQbGa eyypa0JaaGimaaqaaiabg6HiLcqcfaOaeyyeIuoadaWcaaqaaiabeI 7aXnaaCaaabeqcfasaaiabeo7aNjabgUcaRiaadQgaaaqcfa4aaeWa aeaacaaIXaGaey4kaSIaeqiUdehacaGLOaGaayzkaaWaaWbaaeqaju aibaGaeq4SdCMaeyOeI0IaamOAaaaaaKqbagaadaqadaqaaiabeI7a XjabgUcaRiaaikdaaiaawIcacaGLPaaadaahaaqabKqbGeaacqaHZo WzaaaaaKqbaoaapehabaGaamyzamaaCaaabeqcfasaaiabgkHiTiab eI7aXjaaykW7cqaHZoWzcaaMc8UaamiEaaaaaeaacaaIWaaabaGaey OhIukajuaGcqGHRiI8aiaadIhadaahaaqabKqbGeaacaWGQbGaey4k aSIaaGymaiabgkHiTiaaigdaaaqcfaOaamizaiaadIhaaiaawUfaca GLDbaaaaa@78E9@

= 1 1γ log[ j=0 ( γ j ) θ γ+j ( 1+θ ) γj ( θ+2 ) γ Γ( j+1 ) ( θγ ) j+1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacaaIXaaabaGaaGymaiabgkHiTiabeo7aNbaaciGGSbGa ai4BaiaacEgadaWadaqaamaaqahabaWaaeWaaeaafaqabeGabaaaba Gaeq4SdCgabaGaamOAaaaaaiaawIcacaGLPaaaaKqbGeaacaWGQbGa eyypa0JaaGimaaqaaiabg6HiLcqcfaOaeyyeIuoadaWcaaqaaiabeI 7aXnaaCaaabeqcfasaaiabeo7aNjabgUcaRiaadQgaaaqcfa4aaeWa aeaacaaIXaGaey4kaSIaeqiUdehacaGLOaGaayzkaaWaaWbaaeqaju aibaGaeq4SdCMaeyOeI0IaamOAaaaaaKqbagaadaqadaqaaiabeI7a XjabgUcaRiaaikdaaiaawIcacaGLPaaadaahaaqabKqbGeaacqaHZo WzaaaaaKqbaoaalaaabaGaeu4KdC0aaeWaaeaacaWGQbGaey4kaSIa aGymaaGaayjkaiaawMcaaaqaamaabmaabaGaeqiUdeNaeq4SdCgaca GLOaGaayzkaaWaaWbaaeqajuaibaGaamOAaiabgUcaRiaaigdaaaaa aaqcfaOaay5waiaaw2faaaaa@7030@

= 1 1γ log[ j=0 ( γ j ) θ γ1 ( θ+2 ) γ Γ( j+1 ) ( γ ) j+1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaacaaIXaaabaGaaGymaiabgkHiTiabeo7aNbaaciGGSbGa ai4BaiaacEgadaWadaqaamaaqahabaWaaeWaaeaafaqabeGabaaaba Gaeq4SdCgabaGaamOAaaaaaiaawIcacaGLPaaaaKqbGeaacaWGQbGa eyypa0JaaGimaaqaaiabg6HiLcqcfaOaeyyeIuoadaWcaaqaaiabeI 7aXnaaCaaabeqcfasaaiabeo7aNjabgkHiTiaaigdaaaaajuaGbaWa aeWaaeaacqaH4oqCcqGHRaWkcaaIYaaacaGLOaGaayzkaaWaaWbaae qajuaibaGaeq4SdCgaaaaajuaGdaWcaaqaaiabfo5ahnaabmaabaGa amOAaiabgUcaRiaaigdaaiaawIcacaGLPaaaaeaadaqadaqaaiabeo 7aNbGaayjkaiaawMcaamaaCaaabeqcfasaaiaadQgacqGHRaWkcaaI XaaaaaaaaKqbakaawUfacaGLDbaaaaa@6514@

Stress-Strength Reliability

The stress- strength reliability describes the life of a component which has random strength X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwaa aa@3761@ that is subjected to a random stress Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamywaa aa@3762@ . When the stress applied to it exceeds the strength, the component fails instantly and the component will function satisfactorily till X>Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abg6da+iaadMfaaaa@3947@ . Therefore, R=P( Y<X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOuai abg2da9iaadcfadaqadaqaaiaadMfacqGH8aapcaWGybaacaGLOaGa ayzkaaaaaa@3D7E@ is a measure of component reliability and in statistical literature it is known as stress-strength parameter. It has wide applications in almost all areas of knowledge especially in engineering such as structures, deterioration of rocket motors, static fatigue of ceramic components, aging of concrete pressure vessels etc.

Let X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwaa aa@3761@ and Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamywaa aa@3762@ be independent strength and stress random variables having Garima distribution (2.1) with parameter θ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde 3aaSbaaKqbGeaacaaIXaaajuaGbeaaaaa@39D2@  and θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde 3aaSbaaKqbGeaacaaIYaaajuaGbeaaaaa@39D3@  respectively. Then the stress-strength reliability R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOuaa aa@375B@ of Garima distribution can be obtained as

R=P( Y<X )= 0 P( Y<X|X=x ) f X ( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOuai abg2da9iaadcfadaqadaqaaiaadMfacqGH8aapcaWGybaacaGLOaGa ayzkaaGaeyypa0Zaa8qCaeaacaWGqbWaaeWaaeaacaWGzbGaeyipaW JaamiwaiaacYhacaWGybGaeyypa0JaamiEaaGaayjkaiaawMcaaaqc fasaaiaaicdaaeaacqGHEisPaKqbakabgUIiYdGaamOzamaaBaaaju aibaGaamiwaaqcfayabaWaaeWaaeaacaWG4baacaGLOaGaayzkaaGa amizaiaadIhaaaa@53CA@

= 0 f 7 ( x; θ 1 ) F 7 ( x; θ 2 )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 Zaa8qCaeaacaWGMbWaaSbaaKqbGeaacaaI3aaajuaGbeaadaqadaqa aiaadIhacaGG7aGaeqiUde3aaSbaaKqbGeaacaaIXaaajuaGbeaaai aawIcacaGLPaaaaKqbGeaacaaIWaaabaGaeyOhIukajuaGcqGHRiI8 aiaaykW7caaMc8UaamOramaaBaaajuaibaGaaG4naaqcfayabaWaae WaaeaacaWG4bGaai4oaiabeI7aXnaaBaaajuaibaGaaGOmaaqcfaya baaacaGLOaGaayzkaaGaamizaiaadIhaaaa@53D7@

=1 θ 1 [ ( θ 1 + θ 2 ) 2 ( θ 1 θ 2 +2 θ 1 + θ 2 +1 )+2 θ 1 θ 2 ( θ 1 + θ 2 )+2 θ 1 θ 2 ] ( θ 1 +2 )( θ 2 +2 ) ( θ 1 + θ 2 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 JaaGymaiabgkHiTmaalaaabaGaeqiUde3aaSbaaKqbGeaacaaIXaaa juaGbeaadaWadaqaamaabmaabaGaeqiUde3aaSbaaKqbGeaacaaIXa aajuaGbeaacqGHRaWkcqaH4oqCdaWgaaqcfasaaiaaikdaaKqbagqa aaGaayjkaiaawMcaamaaCaaabeqcfasaaiaaikdaaaqcfa4aaeWaae aacqaH4oqCdaWgaaqcfasaaiaaigdaaKqbagqaaiaaykW7cqaH4oqC daWgaaqcfasaaiaaikdaaKqbagqaaiabgUcaRiaaikdacqaH4oqCda WgaaqcfasaaiaaigdaaKqbagqaaiabgUcaRiabeI7aXnaaBaaajuai baGaaGOmaaqcfayabaGaey4kaSIaaGymaaGaayjkaiaawMcaaiabgU caRiaaikdacqaH4oqCdaWgaaqcfasaaiaaigdaaKqbagqaaiaaykW7 cqaH4oqCdaWgaaqcfasaaiaaikdaaKqbagqaamaabmaabaGaeqiUde 3aaSbaaKqbGeaacaaIXaaajuaGbeaacqGHRaWkcqaH4oqCdaWgaaqc fasaaiaaikdaaKqbagqaaaGaayjkaiaawMcaaiabgUcaRiaaikdacq aH4oqCdaWgaaqcfasaaiaaigdaaKqbagqaaiaaykW7cqaH4oqCdaWg aaqcfasaaiaaikdaaKqbagqaaaGaay5waiaaw2faaaqaamaabmaaba GaeqiUde3aaSbaaKqbGeaacaaIXaaajuaGbeaacqGHRaWkcaaIYaaa caGLOaGaayzkaaWaaeWaaeaacqaH4oqCdaWgaaqcfasaaiaaikdaaK qbagqaaiabgUcaRiaaikdaaiaawIcacaGLPaaadaqadaqaaiabeI7a XnaaBaaajuaibaGaaGymaaqcfayabaGaey4kaSIaeqiUde3aaSbaaK qbGeaacaaIYaaajuaGbeaaaiaawIcacaGLPaaadaahaaqabKqbGeaa caaIZaaaaaaaaaa@9123@ .

Estimation of Parameter

Maximum likelihood estimates (MLE)

Let ( x 1 , x 2 , x 3 ,..., x n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWG4bWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaaGPaVlaa dIhadaWgaaqcfasaaiaaikdaaKqbagqaaiaacYcacaaMc8UaamiEam aaBaaajuaibaGaaG4maaqcfayabaGaaiilaiaaykW7caaMc8UaaiOl aiaac6cacaGGUaGaaGPaVlaaykW7caGGSaGaamiEamaaBaaajuaiba GaamOBaaqcfayabaaacaGLOaGaayzkaaaaaa@50B4@  be a random sample from Garima distribution (2.1). The likelihood function, L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitaa aa@3755@ of (2.1) is given by

L= ( θ θ+2 ) n i=1 n ( 1+θ+θ x i ) e nθ x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitai abg2da9maabmaabaWaaSaaaeaacqaH4oqCaeaacqaH4oqCcqGHRaWk caaIYaaaaaGaayjkaiaawMcaamaaCaaabeqcfasaaiaad6gaaaqcfa 4aaebCaeaadaqadaqaaiaaigdacqGHRaWkcqaH4oqCcqGHRaWkcqaH 4oqCcaaMc8UaamiEamaaBaaajuaibaGaamyAaaqcfayabaaacaGLOa GaayzkaaaajuaibaGaamyAaiabg2da9iaaigdaaeaacaWGUbaajuaG cqGHpis1aiaaykW7caWGLbWaaWbaaeqajuaibaGaeyOeI0IaamOBai aaykW7cqaH4oqCcaaMc8UabmiEayaaraaaaaaa@5D93@

The natural log likelihood function is thus obtained as

lnL=nln( θ θ+2 )+ i=1 n ln( 1+θ+θ x i ) nθ x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaciiBai aac6gacaWGmbGaeyypa0JaamOBaiGacYgacaGGUbWaaeWaaeaadaWc aaqaaiabeI7aXbqaaiabeI7aXjabgUcaRiaaikdaaaaacaGLOaGaay zkaaGaey4kaSYaaabCaeaaciGGSbGaaiOBamaabmaabaGaaGymaiab gUcaRiabeI7aXjabgUcaRiabeI7aXjaaykW7caWG4bWaaSbaaKqbGe aacaWGPbaajuaGbeaaaiaawIcacaGLPaaaaKqbGeaacaWGPbGaeyyp a0JaaGymaaqaaiaad6gaaKqbakabggHiLdGaeyOeI0IaamOBaiaayk W7cqaH4oqCcaaMc8UabmiEayaaraaaaa@608F@

Now        dlnL dθ = 2n θ 2 +2θ + i=1 n 1+ x i 1+θ+θ x i n x ¯ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGKbGaciiBaiaac6gacaWGmbaabaGaamizaiabeI7aXbaacqGH 9aqpdaWcaaqaaiaaikdacaWGUbaabaGaeqiUde3aaWbaaeqajuaiba GaaGOmaaaajuaGcqGHRaWkcaaIYaGaeqiUdehaaiabgUcaRmaaqaha baWaaSaaaeaacaaIXaGaey4kaSIaamiEamaaBaaajuaibaGaamyAaa qcfayabaaabaGaaGymaiabgUcaRiabeI7aXjabgUcaRiabeI7aXjaa ykW7caWG4bWaaSbaaKqbGeaacaWGPbaajuaGbeaaaaGaeyOeI0Iaam OBaiaaykW7ceWG4bGbaebaaKqbGeaacaWGPbGaeyypa0JaaGymaaqa aiaad6gaaKqbakabggHiLdGaeyypa0JaaGimaaaa@6297@

where x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaara aaaa@370B@ is the sample mean.

The maximum likelihood estimate, θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@  of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  is the solution of the equation dlogL dθ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGKbGaciiBaiaac+gacaGGNbGaamitaaqaaiaadsgacqaH4oqC aaGaeyypa0JaaGimaaaa@3F7D@  and so it can be obtained by solving the following non-linear equation

i=1 n 1+ x i 1+θ+θ x i + 2n θ 2 +2θ n x ¯ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaabCae aadaWcaaqaaiaaigdacqGHRaWkcaWG4bWaaSbaaKqbGeaacaWGPbaa juaGbeaaaeaacaaIXaGaey4kaSIaeqiUdeNaey4kaSIaeqiUdeNaaG PaVlaadIhadaWgaaqcfasaaiaadMgaaKqbagqaaaaaaKqbGeaacaWG PbGaeyypa0JaaGymaaqaaiaad6gaaKqbakabggHiLdGaey4kaSYaaS aaaeaacaaIYaGaamOBaaqaaiabeI7aXnaaCaaabeqcfasaaiaaikda aaqcfaOaey4kaSIaaGOmaiabeI7aXbaacqGHsislcaWGUbGaaGPaVl qadIhagaqeaiabg2da9iaaicdaaaa@5B44@                                          (12.1.1)

Method of moment estimates (MOME)

Equating the population mean of the Garima distribution to the corresponding sample mean, the method of moment estimate (MOME) θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaGaaaaa@3849@ , of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ can be obtained as

θ ˜ = ( 12 x ¯ )+ 4 x ¯ 2 +8 x ¯ +1 2 x ¯ ; x ¯ >0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaGaacqGH9aqpdaWcaaqaamaabmaabaGaaGymaiabgkHiTiaaikda ceWG4bGbaebaaiaawIcacaGLPaaacqGHRaWkdaGcaaqaaiaaisdace WG4bGbaebadaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaaiIda ceWG4bGbaebacqGHRaWkcaaIXaaabeaaaeaacaaIYaGabmiEayaara aaaiaaykW7caaMc8Uaai4oaiqadIhagaqeaiabg6da+iaaicdaaaa@4F93@                  (12.2.1)

A Numerical Example

In this section the goodness of fit of the Garima distribution has been discussed with an example from behavioral science. The data is related with behavioral sciences, collected by Balakrishnan N et al. [9]. The scale “General Rating of Affective Symptoms for Preschoolers (GRASP)” measures behavioral and emotional problems of children, which can be classified with depressive condition or not according to this scale. A study conducted by the authors in a city located at the south part of Chile has allowed collecting real data corresponding to the scores of the GRASP scale of children with frequency in parenthesis, which are:

19(6)

20(15)

21(14)

22(9)

23(12)

24(10)

25(6)

26(9)

27(8)

28(5)

29(6)

30(4)

31(3)

32(4)

33

34

35(4)

36(2)

37(2)

39

42

44

 

In order to compare distributions, 2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyOeI0 IaaGOmaiGacYgacaGGUbGaamitaaaa@3AE2@ , AIC (Akaike Information Criterion), AICC (Akaike Information Criterion Corrected), BIC (Bayesian Information Criterion),K-S Statistics ( Kolmogorov-Smirnov Statistics)  for above data set have been computed and presented in Table 3.  The formulae for computing AIC, AICC, and BIC are as follows:

AIC=2lnL+2k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyqai aadMeacaWGdbGaeyypa0JaeyOeI0IaaGOmaiGacYgacaGGUbGaamit aiabgUcaRiaaikdacaWGRbaaaa@40D2@ ,    AICC=AIC+ 2k( k+1 ) ( nk1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyqai aadMeacaWGdbGaam4qaiabg2da9iaadgeacaWGjbGaam4qaiabgUca RmaalaaabaGaaGOmaiaadUgadaqadaqaaiaadUgacqGHRaWkcaaIXa aacaGLOaGaayzkaaaabaWaaeWaaeaacaWGUbGaeyOeI0Iaam4Aaiab gkHiTiaaigdaaiaawIcacaGLPaaaaaaaaa@49BF@ ,    BIC=2lnL+klnn MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOqai aadMeacaWGdbGaeyypa0JaeyOeI0IaaGOmaiGacYgacaGGUbGaamit aiabgUcaRiaadUgaciGGSbGaaiOBaiaaykW7caWGUbaaaa@4479@  

The best distribution is the distribution which corresponds to lower values of 2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyOeI0 IaaGOmaiGacYgacaGGUbGaamitaaaa@3AE2@ , AIC, AICC, and BIC.

It can be easily seen from above table that the Garima distribution is better than Aradhana, Sujatha, Akash, Shanker, Lindley and exponential distributions  for modeling behavioral science data and thus Garima distribution should be preferred over Aradhana, Sujatha, Akash, Shanker, Lindley and exponential distributions  for modeling behavioral science data.

Model

ML Estimate

2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyOeI0 IaaGOmaiGacYgacaGGUbGaamitaaaa@3AE2@

AIC

AICC

BIC

Garima

0.05317

188.32

190.32

190.35

193.23

Aradhana

0.11557

989.49

991.49

991.52

994.40

Sujatha

0.11745

985.69

987.69

987.72

990.60

Akash

0.11961

981.28

983.28

983.31

986.18

Shanker

0.07974

1033.10

1035.10

1035.13

1037.99

Lindley

0.07725

1041.64

1043.64

1043.68

1046.54

Exponential

0.04006

1130.26

1132.26

1132.29

1135.16

Table 3: MLE’s,-2ln L, AIC, AICC, and BIC of Garima, Aradhana, Sujatha [4], Akash [2], Shanker [1], Lindley [5] and exponential distributions.

Conclusion

 A one parameter lifetime distribution named, “Garima distribution” has been proposed and studied. Its mathematical properties including shape, moments, skewness, kurtosis, hazard rate function, mean residual life function, stochastic ordering, mean deviations, order statistics, Bonferroni and Lorenz curves, Renyi entropy and stress-strength reliability  have been discussed. The condition under which Garima distribution is over-dispersed, equi-dispersed, and under-dispersed are presented along with the conditions under which Sujatha, Aradhana, Akash, Shanker, Lindley and exponential distributions are over-dispersed, equi-dispersed and under-dispersed. The method of moments and the method of maximum likelihood estimation have also been discussed for estimating its parameter. Finally, a numerical example from behavioral science has been considered for the goodness of fit of Garima distribution and the fit has been compared with Sujatha, Aradhana, Akash, Shanker, Lindley and exponential distributions. The goodness of fit of the Garima distribution shows that it is an important model for modeling behavioral science data. 

NOTE: The paper is named in loving memory of my niece Garima Satypriya, daughter of my respected brother Prof. Uma Shanker, Department of Mathematics, K.K College of Engineering & Management, Biharsharif, Nalanda, India.

References

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