eISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 7 Issue 6

Modeling lifetime data with Weibull-Lindley distribution

Ieren TG, Oyamakin SO, Chukwu AU
Department of Statistics, University of Ibadan, Nigeria
Received: October 15, 2018 | Published: November 23, 2018

Correspondence: Ieren TG, Department of Statistics, University of Ibadan, Ibadan, Nigeria

Citation: Ieren TG, Oyamakin SO, Chukwu AU. Modeling lifetime data with Weibull-Lindley distribution. Biom Biostat Int J. 2018;7(6):532‒544. DOI: 10.15406/bbij.2018.07.00256

Abstract

In this paper a new extension of the Lindley distribution is presented using the Weibull link function introduced and studied by Tahir et al.,1 to develop a Weibull-Lindley distribution. We derive and discuss the mathematical and Statistical properties of the subject distribution along with its reliability analysis and inference for the parameters. Finally, the Weibull-Lindley distribution has been used to model four lifetime datasets and the results show that the proposed generalization performs better than the other known extensions of the Lindley distribution considered for the study.

Keywords: Lindley distribution, Weibull-Lindley distribution, mathematical properties, reliability function, parameter estimation, applications.

Introduction

The Lindley distribution introduced by Lindley et al.,2 in the context of Bayesian analysis as a counter example of fiducial statistics, is defined by its probability density function (PDF) and cumulative distribution function (CDF) as

G( x )=1[ 1+ θx θ+1 ] e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEeadaqada qaaiaadIhaaiaawIcacaGLPaaacqGH9aqpcaaIXaGaeyOeI0YaamWa aeaacaaIXaGaey4kaSYaaSaaaeaacqaH4oqCcaWG4baabaGaeqiUde Naey4kaSIaaGymaaaaaiaawUfacaGLDbaadaqfGaqabSqabeaacqGH sislcqaH4oqCcaWG4baaneaacaWGLbaaaaaa@4CB0@ (1.1)

And

g( x )= θ 2 θ+1 ( 1+x ) e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaam4zamaabm aabaGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiUde3a aWbaaSqabeaacaaIYaaaaaGcbaGaeqiUdeNaey4kaSIaaGymaaaada qadaqaaiaaigdacqGHRaWkcaWG4baacaGLOaGaayzkaaWaaubiaeqa leqabaGaeyOeI0IaeqiUdeNaamiEaaqdbaGaamyzaaaaaaa@4AFE@ (1.2)

respectively. For x>0,θ>0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg6 da+iaaicdacaGGSaGaeqiUdeNaeyOpa4JaaGimaiaacYcaaaa@3F0E@ where θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@392D@ is the scale parameter of the Lindley distribution.

Details of this distribution, its mathematical and statistical properties, estimation of its parameter and application including the superiority of Lindley distribution over exponential distribution has been done by Ghitany et al.,3 We have so many generalized families of distributions proposed by different researchers that are used in extending other distributions to produce compound distributions with better performance. These are several ways of adding one or more parameters to a distribution function which makes the resulting distribution richer and more flexible for modeling data. A brief summary of some of these methods or families of distribution include the beta generalized family (Beta-G) by Eugene et al.,4 the Kumaraswamy-G by Cordeiro et al.5  Transmuted family of distributions by Shaw et al.,6 Gamma-G (type 1) by Zografos et al.,7 McDonald-G by Alexander et al.,8 Gamma-G (type 2) by Risti et al.,9 Gamma-G (type 3) by Torabi et al.,10 Log-gamma-G by Amini et al.,9 Exponentiated T-X by Alzaghal et al.,12 Exponentiated-G (EG) by Cordeiro et al.,13 Logistic-G by Torabi et al.,14 Gamma-X by Alzaatreh et al.,15 Logistic-X by Tahir et al.,16 Weibull-X by Alzaatreh et al.,17 Weibull-G by Bourguignon et al.,18 a new Weibull-G family by Tahir et al.,1 a Lomax-G family by Cordeiro et al.,19 a new generalized Weibull-G family by Cordeiro et al.,20 and Beta Marshall-Olkin family of distributions by Alizadeh et al.,21 and some other families of the distributions.

Hence, there are also some generalizations of the Lindley distribution recently proposed in the literature such as the transmuted Lindley distribution by Merovci et al.,23 the exponentiated Power Lindley distribution by Ashour et al.,24 Generalized Lindley distribution by Nadarajah et al.,24 Transmuted Generalized Lindley distribution by Elgarhy et al.,25 Extended Power Lindley distribution by Alkarni et al.,26 a two-parameter Lindley distribution by Shanker et al.,27 the Lomax-Lindley distribution by Yahaya et al.,28 Transmuted Two-Parameter Lindley distribution by Al-khazaleh et al.,29 and a three-parameter Lindley distribution by Shanker et al.,30 The aim of this article is to introduce a new continuous distribution called Weibull-Lindley distribution (WLnD) from the proposed family by Tahir et al.,1 The remaining parts of this article are presented in sections as follows: We defined the new distribution and give its plots in section 2.1. Section 2.2 derived some properties of the new distribution. Section 2.3 proposes some reliability functions of the new distribution. The order statistics for the new distribution are also given in section 2.4. The maximum likelihood estimates (MLEs) of the unknown model parameters of the new distribution are obtained in section 2.5. In section 3 we carryout application of the proposed model with others to four lifetime datasets. Lastly, in section 4, we give the summary of our work and concluding remarks.

Materials and methods

Construction of Weibull-Lindley distribution (WLnD)

In the next section, we have defined the cdf and pdf of the Weibull-Lindley distribution (WLnD) using the method proposed by Tahir et al.1 According to Tahir et al.,1  the formula or Weibull link function for deriving the cdf and pdf of any Weibull-based continuous distribution is defined as:

F(x)= 0 log[ G(x) ] αβ t β1 e α t β dt = e α { log[ G(x) ] } β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamOraiaacI cacaWG4bGaaiykaiabg2da9maapehabaGaeqySdeMaeqOSdiMaamiD amaaCaaaleqabaGaeqOSdiMaeyOeI0IaaGymaaaakmaavacabeWcbe qaaiabgkHiTiabeg7aHjaadshadaahaaadbeqaaiabek7aIbaaa0qa aiaadwgaaaGccaWGKbGaamiDaaWcbaGaaGimaaqaaiabgkHiTiGacY gacaGGVbGaai4zamaadmaabaGaam4raiaacIcacaWG4bGaaiykaaGa ay5waiaaw2faaaqdcqGHRiI8aOGaeyypa0ZaaubiaeqaleqabaGaey OeI0IaeqySde2aaiWaaeaacqGHsislciGGSbGaai4BaiaacEgadaWa daqaaiaadEeacaGGOaGaamiEaiaacMcaaiaawUfacaGLDbaaaiaawU hacaGL9baadaahaaadbeqaaiabek7aIbaaa0qaaiaadwgaaaaaaa@6978@  (2.1.1)

And

f(x)=αβ g(x) G(x) { log[ G(x) ] } β1 e α { log[ G(x) ] } β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacaWG4bGaaiykaiabg2da9iabeg7aHjabek7aInaalaaabaGaam4z aiaacIcacaWG4bGaaiykaaqaaiaadEeacaGGOaGaamiEaiaacMcaaa WaaiWaaeaacqGHsislciGGSbGaai4BaiaacEgadaWadaqaaiaadEea caGGOaGaamiEaiaacMcaaiaawUfacaGLDbaaaiaawUhacaGL9baada ahaaWcbeqaaiabek7aIjabgkHiTiaaigdaaaGcdaqfGaqabSqabeaa cqGHsislcqaHXoqydaGadaqaaiabgkHiTiGacYgacaGGVbGaai4zam aadmaabaGaam4raiaacIcacaWG4bGaaiykaaGaay5waiaaw2faaaGa ay5Eaiaaw2haamaaCaaameqabaGaeqOSdigaaaqdbaGaamyzaaaaaa a@6491@  (2.1.2)

respectively, where g( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaam4zamaabm aabaGaamiEaaGaayjkaiaawMcaaaaa@3AE9@  and G( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaam4ramaabm aabaGaamiEaaGaayjkaiaawMcaaaaa@3AC9@  are the pdf and cdf of any continuous distribution to be generalized respectively and α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3916@  and β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3918@  are the two additional new parameters responsible for the shape of the distribution.

Using equation (1.1) and (1.2) in (2.1.1) and (2.1.2) and simplifying, we obtain the cdf and pdf of the Weibull-Lindley random variable X as:

F( x;θ,α,β )= e α { log[ 1[ 1+ θx θ+1 ] e θx ] } β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamOramaabm aabaGaamiEaiaacUdacqaH4oqCcaGGSaGaeqySdeMaaiilaiabek7a IbGaayjkaiaawMcaaiabg2da9maavacabeWcbeqaaiabgkHiTiabeg 7aHnaacmaabaGaeyOeI0IaciiBaiaac+gacaGGNbWaamWaaeaacaaI XaGaeyOeI0YaamWaaeaacaaIXaGaey4kaSYaaSaaaeaacqaH4oqCca WG4baabaGaeqiUdeNaey4kaSIaaGymaaaaaiaawUfacaGLDbaadaqf GaqabWqabeaacqGHsislcqaH4oqCcaWG4baaoeaacaWGLbaaaaWcca GLBbGaayzxaaaacaGL7bGaayzFaaWaaWbaaWqabeaacqaHYoGyaaaa neaacaWGLbaaaaaa@6098@  (2.1.3)

And

f(x;θ,α,β)= αβ θ 2 ( 1+x ) e θx ( θ+1 )[ 1[ 1+ θx θ+1 ] e θx ] { log[ 1[ 1+ θx θ+1 ] e θx ] } β1 e α { log[ 1[ 1+ θx θ+1 ] e θx ] } β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacaWG4bGaai4oaiabeI7aXjaacYcacqaHXoqycaGGSaGaeqOSdiMa aiykaiabg2da9maalaaabaGaeqySdeMaeqOSdiMaeqiUde3aaWbaaS qabeaacaaIYaaaaOWaaeWaaeaacaaIXaGaey4kaSIaamiEaaGaayjk aiaawMcaamaavacabeWcbeqaaiabgkHiTiabeI7aXjaadIhaa0qaai aadwgaaaaakeaadaqadaqaaiabeI7aXjabgUcaRiaaigdaaiaawIca caGLPaaadaWadaqaaiaaigdacqGHsisldaWadaqaaiaaigdacqGHRa WkdaWcaaqaaiabeI7aXjaadIhaaeaacqaH4oqCcqGHRaWkcaaIXaaa aaGaay5waiaaw2faamaavacabeWcbeqaaiabgkHiTiabeI7aXjaadI haa0qaaiaadwgaaaaakiaawUfacaGLDbaaaaWaaiWaaeaacqGHsisl ciGGSbGaai4BaiaacEgadaWadaqaaiaaigdacqGHsisldaWadaqaai aaigdacqGHRaWkdaWcaaqaaiabeI7aXjaadIhaaeaacqaH4oqCcqGH RaWkcaaIXaaaaaGaay5waiaaw2faamaavacabeWcbeqaaiabgkHiTi abeI7aXjaadIhaa0qaaiaadwgaaaaakiaawUfacaGLDbaaaiaawUha caGL9baadaahaaWcbeqaaiabek7aIjabgkHiTiaaigdaaaGcdaqfGa qabSqabeaacqGHsislcqaHXoqydaGadaqaaiabgkHiTiGacYgacaGG VbGaai4zamaadmaabaGaaGymaiabgkHiTmaadmaabaGaaGymaiabgU caRmaalaaabaGaeqiUdeNaamiEaaqaaiabeI7aXjabgUcaRiaaigda aaaacaGLBbGaayzxaaWaaubiaeqameqabaGaeyOeI0IaeqiUdeNaam iEaaGdbaGaamyzaaaaaSGaay5waiaaw2faaaGaay5Eaiaaw2haamaa CaaameqabaGaeqOSdigaaaqdbaGaamyzaaaaaaa@A22A@  (2.1.4)

respectively.

For x>0;θ,α,β>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg6 da+iaaicdacaGG7aGaeqiUdeNaaiilaiabeg7aHjaacYcacqaHYoGy cqGH+aGpcaaIWaaaaa@430D@ ; where θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@392D@  is a scale parameter and α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3916@  and β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3918@  is a shape parameters of the Weibull-Lindley distribution.

The following is a graphical representation of the pdf and cdf of the Weibull-Lindley distribution. Given some values of the parameters α,β&θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaai ilaiabek7aIjaacAcacqaH4oqCaaa@3DC7@ , we provide some possible graphs for the pdf and the cdf of the WLnD as shown in Figure 1&2 below: Figure 1 indicates that the WLnD is a skewed distribution and can take various forms. This means that distribution can be very useful for datasets that are skewed.

Figure 1 Graph of PDF of the WLnD for varrying parameter values.

Figure 2 Graph of CDF of the WLnD for varying parameter values.

From the above cdf plot, the cdf increases when X increases, and approaches 1 when X becomes large, as expected.

Properties

In this section, we defined and discuss some properties of the WLnD distribution.

The Quantile function

This function is derived by inverting the cdf of any given continuous probability distribution. It is used for obtaining some moments like skewness and kurtosis as well as the median and for generation of random variables from the distribution in question. Hyndman et al.,21 defined the quantile function for any distribution in the form  Q(u) = F 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaacbiaeaaaaaa aaa8qacaWFgbWdamaaCaaaleqabaWdbiabgkHiTiaaigdaaaaaaa@3A5D@ (u)  where Q(u) is the quantile function of F(x) for 0 < u <1

Taking F(x) to be the cdf of the Weibull-Lindley distribution and inverting it as above will give us the Quantile function as follows:

F( x )= e α { log[ 1[ 1+ θx θ+1 ] e θx ] } β =u MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamOramaabm aabaGaamiEaaGaayjkaiaawMcaaiabg2da9maavacabeWcbeqaaiab gkHiTiabeg7aHnaacmaabaGaeyOeI0IaciiBaiaac+gacaGGNbWaam WaaeaacaaIXaGaeyOeI0YaamWaaeaacaaIXaGaey4kaSYaaSaaaeaa cqaH4oqCcaWG4baabaGaeqiUdeNaey4kaSIaaGymaaaaaiaawUfaca GLDbaadaqfGaqabWqabeaacqGHsislcqaH4oqCcaWG4baaoeaacaWG LbaaaaWccaGLBbGaayzxaaaacaGL7bGaayzFaaWaaWbaaWqabeaacq aHYoGyaaaaneaacaWGLbaaaOGaeyypa0JaamyDaaaa@5B8D@  (2.2.1)

Simplifying equation (2.2.1) above, we obtain:

  Q( u )= X q =1 1 θ 1 θ W( ( θ+1 )( 1exp{ ( lnu α ) 1 α } ) e θ+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamyuamaabm aabaGaamyDaaGaayjkaiaawMcaaiabg2da9iaadIfadaWgaaWcbaGa amyCaaqabaGccqGH9aqpcqGHsislcaaIXaGaeyOeI0YaaSaaaeaaca aIXaaabaGaeqiUdehaaiabgkHiTmaalaaabaGaaGymaaqaaiabeI7a XbaacaWGxbWaaeWaaeaacqGHsisldaWcaaqaamaabmaabaGaeqiUde Naey4kaSIaaGymaaGaayjkaiaawMcaamaabmaabaGaaGymaiabgkHi TiGacwgacaGG4bGaaiiCamaacmaabaGaeyOeI0YaaeWaaeaacqGHsi sldaWcaaqaaiGacYgacaGGUbGaamyDaaqaaiabeg7aHbaaaiaawIca caGLPaaadaahaaWcbeqaamaaleaameaacaaIXaaabaGaeqySdegaaa aaaOGaay5Eaiaaw2haaaGaayjkaiaawMcaaaqaamaavacabeWcbeqa aiabeI7aXjabgUcaRiaaigdaa0qaaiaadwgaaaaaaaGccaGLOaGaay zkaaaaaa@671F@  (2.2.2)

By using (2.2.2) above, the median of X from the WLnD is simply obtained by setting u=0.5 while random numbers can be generated from WLnD by setting X=Q( u ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamiwaiabg2 da9iaadgfadaqadaqaaiaadwhaaiaawIcacaGLPaaaaaa@3CB3@ , where u is a uniform variate on the unit interval (0,1) and W( . ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaam4vamaabm aabaGaaiOlaaGaayjkaiaawMcaaaaa@3A8E@ represents the negative branch of the Lambert function.

Skewness and kurtosis

The quantile based measures of skewness and kurtosis will employed due to non-existence of the classical measures in some cases. The Bowley’s measure of skewness based on quartiles by Kenney et al.,32 is given as;

< SK= Q( 3 4 )2Q( 1 2 )+Q( 1 4 ) Q( 3 4 )Q( 1 4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaam4uaiaadU eacqGH9aqpdaWcaaqaaiaadgfadaqadaqaamaalaaabaGaaG4maaqa aiaaisdaaaaacaGLOaGaayzkaaGaeyOeI0IaaGOmaiaadgfadaqada qaamaalaaabaGaaGymaaqaaiaaikdaaaaacaGLOaGaayzkaaGaey4k aSIaamyuamaabmaabaWaaSaaaeaacaaIXaaabaGaaGinaaaaaiaawI cacaGLPaaaaeaacaWGrbWaaeWaaeaadaWcaaqaaiaaiodaaeaacaaI 0aaaaaGaayjkaiaawMcaaiabgkHiTiaadgfadaqadaqaamaalaaaba GaaGymaaqaaiaaisdaaaaacaGLOaGaayzkaaaaaaaa@5137@  (2.2.3)

while the Risti et al.,9 kurtosis based on octiles is given by;

KT= Q( 7 8 )Q( 5 8 )Q( 3 8 )+( 1 8 ) Q( 6 8 )Q( 1 4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaam4saiaads facqGH9aqpdaWcaaqaaiaadgfadaqadaqaamaalaaabaGaaG4naaqa aiaaiIdaaaaacaGLOaGaayzkaaGaeyOeI0IaamyuamaabmaabaWaaS aaaeaacaaI1aaabaGaaGioaaaaaiaawIcacaGLPaaacqGHsislcaWG rbWaaeWaaeaadaWcaaqaaiaaiodaaeaacaaI4aaaaaGaayjkaiaawM caaiabgUcaRmaabmaabaWaaSaaaeaacaaIXaaabaGaaGioaaaaaiaa wIcacaGLPaaaaeaacaWGrbWaaeWaaeaadaWcaaqaaiaaiAdaaeaaca aI4aaaaaGaayjkaiaawMcaaiabgkHiTiaadgfadaqadaqaamaalaaa baGaaGymaaqaaiaaisdaaaaacaGLOaGaayzkaaaaaaaa@549E@  (2.2.4)

where Q(.) is any quartile or octile of interest.

Moments

Moments of a random variable are very important in distribution theory because they are used to study some of the most important features and characteristics of a random variable such as mean, variance, skewness and kurtosis.

Let X denote a continuous random variable, the nth moment of X is given by;

μ n ' =E[ X n ]= 0 x n f(x)dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaWaaubmaeqale aacaWGUbaabaGaai4jaaqdbaGaeqiVd0gaaOGaeyypa0Jaamyramaa dmaabaWaaubiaeqaleqabaGaamOBaaqdbaGaamiwaaaaaOGaay5wai aaw2faaiabg2da9maapehabaWaaubiaeqaleqabaGaamOBaaqdbaGa amiEaaaakiaadAgacaGGOaGaamiEaiaacMcacaWGKbGaamiEaaWcba GaaGimaaqaaiabg6HiLcqdcqGHRiI8aaaa@4E3D@ (2.2.5)

Considering f(x) to be the pdf of the Weibull-Lindley distribution as given in equation (2.1.4)

μ n ' =E[ X n ]= 0 1 x n f(x)dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaWaaubmaeqale aacaWGUbaabaGaai4jaaqdbaGaeqiVd0gaaOGaeyypa0Jaamyramaa dmaabaWaaubiaeqaleqabaGaamOBaaqdbaGaamiwaaaaaOGaay5wai aaw2faaiabg2da9maapehabaWaaubiaeqaleqabaGaamOBaaqdbaGa amiEaaaakiaadAgacaGGOaGaamiEaiaacMcacaWGKbGaamiEaaWcba GaaGimaaqaaiaaigdaa0Gaey4kIipaaaa@4D87@

Recall that from equation (2.1.4),

f(x;θ,α,β)= αβ θ 2 ( 1+x ) e θx ( θ+1 )[ 1[ 1+ θx θ+1 ] e θx ] { log[ 1[ 1+ θx θ+1 ] e θx ] } β1 e α { log[ 1[ 1+ θx θ+1 ] e θx ] } β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacaWG4bGaai4oaiabeI7aXjaacYcacqaHXoqycaGGSaGaeqOSdiMa aiykaiabg2da9maalaaabaGaeqySdeMaeqOSdiMaeqiUde3aaWbaaS qabeaacaaIYaaaaOWaaeWaaeaacaaIXaGaey4kaSIaamiEaaGaayjk aiaawMcaamaavacabeWcbeqaaiabgkHiTiabeI7aXjaadIhaa0qaai aadwgaaaaakeaadaqadaqaaiabeI7aXjabgUcaRiaaigdaaiaawIca caGLPaaadaWadaqaaiaaigdacqGHsisldaWadaqaaiaaigdacqGHRa WkdaWcaaqaaiabeI7aXjaadIhaaeaacqaH4oqCcqGHRaWkcaaIXaaa aaGaay5waiaaw2faamaavacabeWcbeqaaiabgkHiTiabeI7aXjaadI haa0qaaiaadwgaaaaakiaawUfacaGLDbaaaaWaaiWaaeaacqGHsisl ciGGSbGaai4BaiaacEgadaWadaqaaiaaigdacqGHsisldaWadaqaai aaigdacqGHRaWkdaWcaaqaaiabeI7aXjaadIhaaeaacqaH4oqCcqGH RaWkcaaIXaaaaaGaay5waiaaw2faamaavacabeWcbeqaaiabgkHiTi abeI7aXjaadIhaa0qaaiaadwgaaaaakiaawUfacaGLDbaaaiaawUha caGL9baadaahaaWcbeqaaiabek7aIjabgkHiTiaaigdaaaGcdaqfGa qabSqabeaacqGHsislcqaHXoqydaGadaqaaiabgkHiTiGacYgacaGG VbGaai4zamaadmaabaGaaGymaiabgkHiTmaadmaabaGaaGymaiabgU caRmaalaaabaGaeqiUdeNaamiEaaqaaiabeI7aXjabgUcaRiaaigda aaaacaGLBbGaayzxaaWaaubiaeqameqabaGaeyOeI0IaeqiUdeNaam iEaaGdbaGaamyzaaaaaSGaay5waiaaw2faaaGaay5Eaiaaw2haamaa CaaameqabaGaeqOSdigaaaqdbaGaamyzaaaaaaa@A22A@  (2.2.6)

Let

A= e α { log[ 1[ 1+ θx θ+1 ] e θx ] } β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2 da9maavacabeWcbeqaaiabgkHiTiabeg7aHnaacmaabaGaeyOeI0Ia ciiBaiaac+gacaGGNbWaamWaaeaacaaIXaGaeyOeI0YaamWaaeaaca aIXaGaey4kaSYaaSaaaeaacqaH4oqCcaWG4baabaGaeqiUdeNaey4k aSIaaGymaaaaaiaawUfacaGLDbaadaqfGaqabWqabeaacqGHsislcq aH4oqCcaWG4baaoeaacaWGLbaaaaWccaGLBbGaayzxaaaacaGL7bGa ayzFaaWaaWbaaWqabeaacqaHYoGyaaaaneaacaWGLbaaaaaa@56F8@

Then, using a power series expansion for A, we can write A as:

A= e α { log[ 1[ 1+ θx θ+1 ] e θx ] } β = i=0 ( 1 ) i α i i! ( log[ 1[ 1+ θx θ+1 ] e θx ] ) βi MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2 da9maavacabeWcbeqaaiabgkHiTiabeg7aHnaacmaabaGaeyOeI0Ia ciiBaiaac+gacaGGNbWaamWaaeaacaaIXaGaeyOeI0YaamWaaeaaca aIXaGaey4kaSYaaSaaaeaacqaH4oqCcaWG4baabaGaeqiUdeNaey4k aSIaaGymaaaaaiaawUfacaGLDbaadaqfGaqabWqabeaacqGHsislcq aH4oqCcaWG4baaoeaacaWGLbaaaaWccaGLBbGaayzxaaaacaGL7bGa ayzFaaWaaWbaaWqabeaacqaHYoGyaaaaneaacaWGLbaaaOGaeyypa0 ZaaabCaeaadaWcaaqaamaabmaabaGaeyOeI0IaaGymaaGaayjkaiaa wMcaamaaCaaaleqabaGaamyAaaaakiabeg7aHnaaCaaaleqabaGaam yAaaaaaOqaaiaadMgacaGGHaaaaaWcbaGaamyAaiabg2da9iaaicda aeaacqGHEisPa0GaeyyeIuoakmaabmaabaGaeyOeI0IaciiBaiaac+ gacaGGNbWaamWaaeaacaaIXaGaeyOeI0YaamWaaeaacaaIXaGaey4k aSYaaSaaaeaacqaH4oqCcaWG4baabaGaeqiUdeNaey4kaSIaaGymaa aaaiaawUfacaGLDbaadaqfGaqabSqabeaacqGHsislcqaH4oqCcaWG 4baaneaacaWGLbaaaaGccaGLBbGaayzxaaaacaGLOaGaayzkaaWaaW baaSqabeaacqaHYoGycaWGPbaaaaaa@815B@

Substituting for the expansion above in equation (2.2.6), we have;

f(x)= αβ θ 2 ( 1+x ) e θx ( θ+1 )[ 1[ 1+ θx θ+1 ] e θx ] { log[ 1[ 1+ θx θ+1 ] e θx ] } β1 i=0 ( 1 ) i α i i! ( log[ 1[ 1+ θx θ+1 ] e θx ] ) βi MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacaWG4bGaaiykaiabg2da9maalaaabaGaeqySdeMaeqOSdiMaeqiU de3aaWbaaSqabeaacaaIYaaaaOWaaeWaaeaacaaIXaGaey4kaSIaam iEaaGaayjkaiaawMcaamaavacabeWcbeqaaiabgkHiTiabeI7aXjaa dIhaa0qaaiaadwgaaaaakeaadaqadaqaaiabeI7aXjabgUcaRiaaig daaiaawIcacaGLPaaadaWadaqaaiaaigdacqGHsisldaWadaqaaiaa igdacqGHRaWkdaWcaaqaaiabeI7aXjaadIhaaeaacqaH4oqCcqGHRa WkcaaIXaaaaaGaay5waiaaw2faamaavacabeWcbeqaaiabgkHiTiab eI7aXjaadIhaa0qaaiaadwgaaaaakiaawUfacaGLDbaaaaWaaiWaae aacqGHsislciGGSbGaai4BaiaacEgadaWadaqaaiaaigdacqGHsisl daWadaqaaiaaigdacqGHRaWkdaWcaaqaaiabeI7aXjaadIhaaeaacq aH4oqCcqGHRaWkcaaIXaaaaaGaay5waiaaw2faamaavacabeWcbeqa aiabgkHiTiabeI7aXjaadIhaa0qaaiaadwgaaaaakiaawUfacaGLDb aaaiaawUhacaGL9baadaahaaWcbeqaaiabek7aIjabgkHiTiaaigda aaGcdaaeWbqaamaalaaabaWaaeWaaeaacqGHsislcaaIXaaacaGLOa GaayzkaaWaaWbaaSqabeaacaWGPbaaaOGaeqySde2aaWbaaSqabeaa caWGPbaaaaGcbaGaamyAaiaacgcaaaaaleaacaWGPbGaeyypa0JaaG imaaqaaiabg6HiLcqdcqGHris5aOWaaeWaaeaacqGHsislciGGSbGa ai4BaiaacEgadaWadaqaaiaaigdacqGHsisldaWadaqaaiaaigdacq GHRaWkdaWcaaqaaiabeI7aXjaadIhaaeaacqaH4oqCcqGHRaWkcaaI XaaaaaGaay5waiaaw2faamaavacabeWcbeqaaiabgkHiTiabeI7aXj aadIhaa0qaaiaadwgaaaaakiaawUfacaGLDbaaaiaawIcacaGLPaaa daahaaWcbeqaaiabek7aIjaadMgaaaaaaa@A6B3@

f( x )= αβ θ 2 ( 1+x ) e θx ( θ+1 )[ 1[ 1+ θx θ+1 ] e θx ] i=0 ( 1 ) i α i i! ( log[ 1[ 1+ θx θ+1 ] e θx ] ) β( i+1 )1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm aabaGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqySdeMa eqOSdiMaeqiUde3aaWbaaSqabeaacaaIYaaaaOWaaeWaaeaacaaIXa Gaey4kaSIaamiEaaGaayjkaiaawMcaamaavacabeWcbeqaaiabgkHi TiabeI7aXjaadIhaa0qaaiaadwgaaaaakeaadaqadaqaaiabeI7aXj abgUcaRiaaigdaaiaawIcacaGLPaaadaWadaqaaiaaigdacqGHsisl daWadaqaaiaaigdacqGHRaWkdaWcaaqaaiabeI7aXjaadIhaaeaacq aH4oqCcqGHRaWkcaaIXaaaaaGaay5waiaaw2faamaavacabeWcbeqa aiabgkHiTiabeI7aXjaadIhaa0qaaiaadwgaaaaakiaawUfacaGLDb aaaaWaaabCaeaadaWcaaqaamaabmaabaGaeyOeI0IaaGymaaGaayjk aiaawMcaamaaCaaaleqabaGaamyAaaaakiabeg7aHnaaCaaaleqaba GaamyAaaaaaOqaaiaadMgacaGGHaaaaaWcbaGaamyAaiabg2da9iaa icdaaeaacqGHEisPa0GaeyyeIuoakmaabmaabaGaeyOeI0IaciiBai aac+gacaGGNbWaamWaaeaacaaIXaGaeyOeI0YaamWaaeaacaaIXaGa ey4kaSYaaSaaaeaacqaH4oqCcaWG4baabaGaeqiUdeNaey4kaSIaaG ymaaaaaiaawUfacaGLDbaadaqfGaqabSqabeaacqGHsislcqaH4oqC caWG4baaneaacaWGLbaaaaGccaGLBbGaayzxaaaacaGLOaGaayzkaa WaaWbaaSqabeaacqaHYoGydaqadaqaaiaadMgacqGHRaWkcaaIXaaa caGLOaGaayzkaaGaeyOeI0IaaGymaaaaaaa@901A@  (JKJK) (2.2.7)

Also, let

B= ( log[ 1[ 1+ θx θ+1 ] e θx ] ) β( i+1 )1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamOqaiabg2 da9maabmaabaGaeyOeI0IaciiBaiaac+gacaGGNbWaamWaaeaacaaI XaGaeyOeI0YaamWaaeaacaaIXaGaey4kaSYaaSaaaeaacqaH4oqCca WG4baabaGaeqiUdeNaey4kaSIaaGymaaaaaiaawUfacaGLDbaadaqf GaqabSqabeaacqGHsislcqaH4oqCcaWG4baaneaacaWGLbaaaaGcca GLBbGaayzxaaaacaGLOaGaayzkaaWaaWbaaSqabeaacqaHYoGydaqa daqaaiaadMgacqGHRaWkcaaIXaaacaGLOaGaayzkaaGaeyOeI0IaaG ymaaaaaaa@583D@

Now, considering the following formula from Tahir et al.,1 which holds for B for i≥1, and then we can write B as follows:

( log[ 1[ 1+ θx θ+1 ] e θx ] ) β( i+1 )1 = k,l=0 j=0 k ( 1 ) j+k+l ( β( i+1 ) ) ( β( i+1 )1j ) ( k( β( i+1 )1 ) k ) ( k j )( ( β( i+1 )1 )+k l ) P j,k [ 1[ 1+ θx θ+1 ] e θx ] l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq GHsislciGGSbGaai4BaiaacEgadaWadaqaaiaaigdacqGHsisldaWa daqaaiaaigdacqGHRaWkdaWcaaqaaiabeI7aXjaadIhaaeaacqaH4o qCcqGHRaWkcaaIXaaaaaGaay5waiaaw2faamaavacabeWcbeqaaiab gkHiTiabeI7aXjaadIhaa0qaaiaadwgaaaaakiaawUfacaGLDbaaai aawIcacaGLPaaadaahaaWcbeqaaiabek7aInaabmaabaGaamyAaiab gUcaRiaaigdaaiaawIcacaGLPaaacqGHsislcaaIXaaaaOGaeyypa0 ZaaabCaeaadaaeWbqaamaalaaabaWaaeWaaeaacqGHsislcaaIXaaa caGLOaGaayzkaaWaaWbaaSqabeaacaWGQbGaey4kaSIaam4AaiabgU caRiaadYgaaaGcdaqadaqaaiabek7aInaabmaabaGaamyAaiabgUca RiaaigdaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaeaadaqadaqaai abek7aInaabmaabaGaamyAaiabgUcaRiaaigdaaiaawIcacaGLPaaa cqGHsislcaaIXaGaeyOeI0IaamOAaaGaayjkaiaawMcaaaaadaqada qaauaabeqaceaaaeaacaWGRbGaeyOeI0YaaeWaaeaacqaHYoGydaqa daqaaiaadMgacqGHRaWkcaaIXaaacaGLOaGaayzkaaGaeyOeI0IaaG ymaaGaayjkaiaawMcaaaqaaiaadUgaaaaacaGLOaGaayzkaaaaleaa caWGQbGaeyypa0JaaGimaaqaaiaadUgaa0GaeyyeIuoaaSqaaiaadU gacaGGSaGaamiBaiabg2da9iaaicdaaeaacqGHEisPa0GaeyyeIuoa kmaabmaabaqbaeqabiqaaaqaaiaadUgaaeaacaWGQbaaaaGaayjkai aawMcaamaabmaabaqbaeqabiqaaaqaamaabmaabaGaeqOSdi2aaeWa aeaacaWGPbGaey4kaSIaaGymaaGaayjkaiaawMcaaiabgkHiTiaaig daaiaawIcacaGLPaaacqGHRaWkcaWGRbaabaGaamiBaaaaaiaawIca caGLPaaacaWGqbWaaSbaaSqaaiaadQgacaGGSaGaam4AaaqabaGcda WadaqaaiaaigdacqGHsisldaWadaqaaiaaigdacqGHRaWkdaWcaaqa aiabeI7aXjaadIhaaeaacqaH4oqCcqGHRaWkcaaIXaaaaaGaay5wai aaw2faamaavacabeWcbeqaaiabgkHiTiabeI7aXjaadIhaa0qaaiaa dwgaaaaakiaawUfacaGLDbaadaahaaWcbeqaaiaadYgaaaaaaa@B459@  (2.2.8)

Where for (for j≥0) Pj,0=1 and (for k=1,2,…..)

P j,k = k 1 m=1 k ( 1 ) m [ m( j+1 )k ] ( m+1 ) P j,km MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGQbGaaiilaiaadUgaaeqaaOGaeyypa0Jaam4AamaaCaaa leqabaGaeyOeI0IaaGymaaaakmaaqahabaWaaeWaaeaacqGHsislca aIXaaacaGLOaGaayzkaaWaaWbaaSqabeaacaWGTbaaaOWaaSaaaeaa daWadaqaaiaad2gadaqadaqaaiaadQgacqGHRaWkcaaIXaaacaGLOa GaayzkaaGaeyOeI0Iaam4AaaGaay5waiaaw2faaaqaamaabmaabaGa amyBaiabgUcaRiaaigdaaiaawIcacaGLPaaaaaGaamiuamaaBaaale aacaWGQbGaaiilaiaadUgacqGHsislcaWGTbaabeaaaeaacaWGTbGa eyypa0JaaGymaaqaaiaadUgaa0GaeyyeIuoaaaa@5B87@  (2.2.9)

Combining equation (2.2.8) and (2.2.9) and inserting the above power series in equation (2.2.7) and simplifying, we have: 

f( x )= αβ θ 2 ( θ+1 ) i=0 ( 1 ) i α i i! k,l=0 j=0 k ( 1 ) j+k+l ( β( i+1 ) ) ( β( i+1 )1j ) ( k( β( i+1 )1 ) k ) ( k j )( ( β( i+1 )1 )+k l ) P j,k ( 1+x ) e θx [ 1[ 1+ θx θ+1 ] e θx ] l1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm aabaGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqySdeMa eqOSdiMaeqiUde3aaWbaaSqabeaacaaIYaaaaaGcbaWaaeWaaeaacq aH4oqCcqGHRaWkcaaIXaaacaGLOaGaayzkaaaaamaaqahabaWaaSaa aeaadaqadaqaaiabgkHiTiaaigdaaiaawIcacaGLPaaadaahaaWcbe qaaiaadMgaaaGccqaHXoqydaahaaWcbeqaaiaadMgaaaaakeaacaWG PbGaaiyiaaaaaSqaaiaadMgacqGH9aqpcaaIWaaabaGaeyOhIukani abggHiLdGcdaaeWbqaamaaqahabaWaaSaaaeaadaqadaqaaiabgkHi TiaaigdaaiaawIcacaGLPaaadaahaaWcbeqaaiaadQgacqGHRaWkca WGRbGaey4kaSIaamiBaaaakmaabmaabaGaeqOSdi2aaeWaaeaacaWG PbGaey4kaSIaaGymaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaqaam aabmaabaGaeqOSdi2aaeWaaeaacaWGPbGaey4kaSIaaGymaaGaayjk aiaawMcaaiabgkHiTiaaigdacqGHsislcaWGQbaacaGLOaGaayzkaa aaamaabmaabaqbaeqabiqaaaqaaiaadUgacqGHsisldaqadaqaaiab ek7aInaabmaabaGaamyAaiabgUcaRiaaigdaaiaawIcacaGLPaaacq GHsislcaaIXaaacaGLOaGaayzkaaaabaGaam4AaaaaaiaawIcacaGL PaaaaSqaaiaadQgacqGH9aqpcaaIWaaabaGaam4AaaqdcqGHris5aa WcbaGaam4AaiaacYcacaWGSbGaeyypa0JaaGimaaqaaiabg6HiLcqd cqGHris5aOWaaeWaaeaafaqabeGabaaabaGaam4AaaqaaiaadQgaaa aacaGLOaGaayzkaaWaaeWaaeaafaqabeGabaaabaWaaeWaaeaacqaH YoGydaqadaqaaiaadMgacqGHRaWkcaaIXaaacaGLOaGaayzkaaGaey OeI0IaaGymaaGaayjkaiaawMcaaiabgUcaRiaadUgaaeaacaWGSbaa aaGaayjkaiaawMcaaiaadcfadaWgaaWcbaGaamOAaiaacYcacaWGRb aabeaakmaabmaabaGaaGymaiabgUcaRiaadIhaaiaawIcacaGLPaaa daqfGaqabSqabeaacqGHsislcqaH4oqCcaWG4baaneaacaWGLbaaaO WaamWaaeaacaaIXaGaeyOeI0YaamWaaeaacaaIXaGaey4kaSYaaSaa aeaacqaH4oqCcaWG4baabaGaeqiUdeNaey4kaSIaaGymaaaaaiaawU facaGLDbaadaqfGaqabSqabeaacqGHsislcqaH4oqCcaWG4baaneaa caWGLbaaaaGccaGLBbGaayzxaaWaaWbaaSqabeaacaWGSbGaeyOeI0 IaaGymaaaaaaa@BD79@

f( x )= αβ θ 2 ( θ+1 ) i=0 k,l=0 j=0 k ( 1 ) i+j+k+l α i+1 ( β( i+1 ) ) i!( β( i+1 )1j ) ( k( β( i+1 )1 ) k ) ( k j )( ( β( i+1 )1 )+k l ) P j,k ( 1+x ) e θx [ 1[ 1+ θx θ+1 ] e θx ] l1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm aabaGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqySdeMa eqOSdiMaeqiUde3aaWbaaSqabeaacaaIYaaaaaGcbaWaaeWaaeaacq aH4oqCcqGHRaWkcaaIXaaacaGLOaGaayzkaaaaamaaqahabaWaaabC aeaadaaeWbqaamaalaaabaWaaeWaaeaacqGHsislcaaIXaaacaGLOa GaayzkaaWaaWbaaSqabeaacaWGPbGaey4kaSIaamOAaiabgUcaRiaa dUgacqGHRaWkcaWGSbaaaOGaeqySde2aaWbaaSqabeaacaWGPbGaey 4kaSIaaGymaaaakmaabmaabaGaeqOSdi2aaeWaaeaacaWGPbGaey4k aSIaaGymaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaqaaiaadMgaca GGHaWaaeWaaeaacqaHYoGydaqadaqaaiaadMgacqGHRaWkcaaIXaaa caGLOaGaayzkaaGaeyOeI0IaaGymaiabgkHiTiaadQgaaiaawIcaca GLPaaaaaWaaeWaaeaafaqabeGabaaabaGaam4AaiabgkHiTmaabmaa baGaeqOSdi2aaeWaaeaacaWGPbGaey4kaSIaaGymaaGaayjkaiaawM caaiabgkHiTiaaigdaaiaawIcacaGLPaaaaeaacaWGRbaaaaGaayjk aiaawMcaaaWcbaGaamOAaiabg2da9iaaicdaaeaacaWGRbaaniabgg HiLdaaleaacaWGRbGaaiilaiaadYgacqGH9aqpcaaIWaaabaGaeyOh IukaniabggHiLdGcdaqadaqaauaabeqaceaaaeaacaWGRbaabaGaam OAaaaaaiaawIcacaGLPaaadaqadaqaauaabeqaceaaaeaadaqadaqa aiabek7aInaabmaabaGaamyAaiabgUcaRiaaigdaaiaawIcacaGLPa aacqGHsislcaaIXaaacaGLOaGaayzkaaGaey4kaSIaam4Aaaqaaiaa dYgaaaaacaGLOaGaayzkaaGaamiuamaaBaaaleaacaWGQbGaaiilai aadUgaaeqaaOWaaeWaaeaacaaIXaGaey4kaSIaamiEaaGaayjkaiaa wMcaamaavacabeWcbeqaaiabgkHiTiabeI7aXjaadIhaa0qaaiaadw gaaaGcdaWadaqaaiaaigdacqGHsisldaWadaqaaiaaigdacqGHRaWk daWcaaqaaiabeI7aXjaadIhaaeaacqaH4oqCcqGHRaWkcaaIXaaaaa Gaay5waiaaw2faamaavacabeWcbeqaaiabgkHiTiabeI7aXjaadIha a0qaaiaadwgaaaaakiaawUfacaGLDbaadaahaaWcbeqaaiaadYgacq GHsislcaaIXaaaaaqaaiaadMgacqGH9aqpcaaIWaaabaGaeyOhIuka niabggHiLdaaaa@BC6B@ (2.2.10)

Now,if l is a positive non-integer, we can expand the last term in (2.2.10) as:

[ 1[ 1+ θx θ+1 ] e θx ] l1 = m=0 ( 1 ) m ( l1 m ) [ [ 1+ θx θ+1 ] e θx ] m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaaca aIXaGaeyOeI0YaamWaaeaacaaIXaGaey4kaSYaaSaaaeaacqaH4oqC caWG4baabaGaeqiUdeNaey4kaSIaaGymaaaaaiaawUfacaGLDbaada qfGaqabSqabeaacqGHsislcqaH4oqCcaWG4baaneaacaWGLbaaaaGc caGLBbGaayzxaaWaaWbaaSqabeaacaWGSbGaeyOeI0IaaGymaaaaki abg2da9maaqahabaWaaeWaaeaacqGHsislcaaIXaaacaGLOaGaayzk aaWaaWbaaSqabeaacaWGTbaaaaqaaiaad2gacqGH9aqpcaaIWaaaba GaeyOhIukaniabggHiLdGcdaqadaqaauaabeqaceaaaeaacaWGSbGa eyOeI0IaaGymaaqaaiaad2gaaaaacaGLOaGaayzkaaWaamWaaeaada WadaqaaiaaigdacqGHRaWkdaWcaaqaaiabeI7aXjaadIhaaeaacqaH 4oqCcqGHRaWkcaaIXaaaaaGaay5waiaaw2faamaavacabeWcbeqaai abgkHiTiabeI7aXjaadIhaa0qaaiaadwgaaaaakiaawUfacaGLDbaa daahaaWcbeqaaiaad2gaaaaaaa@6EEB@  (2.2.11)

Therefore, f(x) becomes:

f( x )= αβ θ 2 ( θ+1 ) m=0 i=0 k,l=0 j=0 k ( 1 ) i+j+k+l+m α i+1 ( β( i+1 ) ) i!( β( i+1 )1j ) ( k( β( i+1 )1 ) k ) ( l1 m )( k j )( ( β( i+1 )1 )+k l ) P j,k ( 1+x ) e θx( 1+m ) [ 1+ θx θ+1 ] m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm aabaGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqySdeMa eqOSdiMaeqiUde3aaWbaaSqabeaacaaIYaaaaaGcbaWaaeWaaeaacq aH4oqCcqGHRaWkcaaIXaaacaGLOaGaayzkaaaaamaaqahabaWaaabC aeaadaaeWbqaamaaqahabaWaaSaaaeaadaqadaqaaiabgkHiTiaaig daaiaawIcacaGLPaaadaahaaWcbeqaaiaadMgacqGHRaWkcaWGQbGa ey4kaSIaam4AaiabgUcaRiaadYgacqGHRaWkcaWGTbaaaOGaeqySde 2aaWbaaSqabeaacaWGPbGaey4kaSIaaGymaaaakmaabmaabaGaeqOS di2aaeWaaeaacaWGPbGaey4kaSIaaGymaaGaayjkaiaawMcaaaGaay jkaiaawMcaaaqaaiaadMgacaGGHaWaaeWaaeaacqaHYoGydaqadaqa aiaadMgacqGHRaWkcaaIXaaacaGLOaGaayzkaaGaeyOeI0IaaGymai abgkHiTiaadQgaaiaawIcacaGLPaaaaaWaaeWaaeaafaqabeGabaaa baGaam4AaiabgkHiTmaabmaabaGaeqOSdi2aaeWaaeaacaWGPbGaey 4kaSIaaGymaaGaayjkaiaawMcaaiabgkHiTiaaigdaaiaawIcacaGL PaaaaeaacaWGRbaaaaGaayjkaiaawMcaaaWcbaGaamOAaiabg2da9i aaicdaaeaacaWGRbaaniabggHiLdaaleaacaWGRbGaaiilaiaadYga cqGH9aqpcaaIWaaabaGaeyOhIukaniabggHiLdGcdaqadaqaauaabe qaceaaaeaacaWGSbGaeyOeI0IaaGymaaqaaiaad2gaaaaacaGLOaGa ayzkaaWaaeWaaeaafaqabeGabaaabaGaam4AaaqaaiaadQgaaaaaca GLOaGaayzkaaWaaeWaaeaafaqabeGabaaabaWaaeWaaeaacqaHYoGy daqadaqaaiaadMgacqGHRaWkcaaIXaaacaGLOaGaayzkaaGaeyOeI0 IaaGymaaGaayjkaiaawMcaaiabgUcaRiaadUgaaeaacaWGSbaaaaGa ayjkaiaawMcaaiaadcfadaWgaaWcbaGaamOAaiaacYcacaWGRbaabe aakmaabmaabaGaaGymaiabgUcaRiaadIhaaiaawIcacaGLPaaadaqf GaqabSqabeaacqGHsislcqaH4oqCcaWG4bWaaeWaaeaacaaIXaGaey 4kaSIaamyBaaGaayjkaiaawMcaaaqdbaGaamyzaaaaaSqaaiaadMga cqGH9aqpcaaIWaaabaGaeyOhIukaniabggHiLdaaleaacaWGTbGaey ypa0JaaGimaaqaaiabg6HiLcqdcqGHris5aOWaamWaaeaacaaIXaGa ey4kaSYaaSaaaeaacqaH4oqCcaWG4baabaGaeqiUdeNaey4kaSIaaG ymaaaaaiaawUfacaGLDbaadaahaaWcbeqaaiaad2gaaaaaaa@C3BD@  (2.2.12)

Using power series expansion on the last term in equation (2.2.12), we have

( 1+ θx θ+1 ) m = p=0 ( m p ) ( θx θ+1 ) p = p=0 ( m p ) ( θ θ+1 ) p x p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIXaGaey4kaSYaaSaaaeaacqaH4oqCcaWG4baabaGaeqiUdeNaey4k aSIaaGymaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaad2gaaaGccq GH9aqpdaaeWbqaamaabmaabaqbaeqabiqaaaqaaiaad2gaaeaacaWG WbaaaaGaayjkaiaawMcaaaWcbaGaamiCaiabg2da9iaaicdaaeaacq GHEisPa0GaeyyeIuoakmaabmaabaWaaSaaaeaacqaH4oqCcaWG4baa baGaeqiUdeNaey4kaSIaaGymaaaaaiaawIcacaGLPaaadaahaaWcbe qaaiaadchaaaGccqGH9aqpdaaeWbqaamaabmaabaqbaeqabiqaaaqa aiaad2gaaeaacaWGWbaaaaGaayjkaiaawMcaaaWcbaGaamiCaiabg2 da9iaaicdaaeaacqGHEisPa0GaeyyeIuoakmaabmaabaWaaSaaaeaa cqaH4oqCaeaacqaH4oqCcqGHRaWkcaaIXaaaaaGaayjkaiaawMcaam aaCaaaleqabaGaamiCaaaakiaadIhadaahaaWcbeqaaiaadchaaaaa aa@6A7E@  (2.2.13)

Now, substituting equation (2.2.13), the power series expansion in equation (2.2.12) above, one gets:

f( x )= αβ θ 2 ( θ+1 ) p=0 m=0 i=0 k,l=0 j=0 k ( 1 ) i+j+k+l+m α i+1 ( β( i+1 ) ) i!( β( i+1 )1j ) ( k( β( i+1 )1 ) k ) ( l1 m )( m p ) ( θ θ+1 ) p ( k j )( ( β( i+1 )1 )+k l ) P j,k ( 1+x ) x p e θx( 1+m ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm aabaGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqySdeMa eqOSdiMaeqiUde3aaWbaaSqabeaacaaIYaaaaaGcbaWaaeWaaeaacq aH4oqCcqGHRaWkcaaIXaaacaGLOaGaayzkaaaaamaaqahabaWaaabC aeaadaaeWbqaamaaqahabaWaaabCaeaadaWcaaqaamaabmaabaGaey OeI0IaaGymaaGaayjkaiaawMcaamaaCaaaleqabaGaamyAaiabgUca RiaadQgacqGHRaWkcaWGRbGaey4kaSIaamiBaiabgUcaRiaad2gaaa GccqaHXoqydaahaaWcbeqaaiaadMgacqGHRaWkcaaIXaaaaOWaaeWa aeaacqaHYoGydaqadaqaaiaadMgacqGHRaWkcaaIXaaacaGLOaGaay zkaaaacaGLOaGaayzkaaaabaGaamyAaiaacgcadaqadaqaaiabek7a InaabmaabaGaamyAaiabgUcaRiaaigdaaiaawIcacaGLPaaacqGHsi slcaaIXaGaeyOeI0IaamOAaaGaayjkaiaawMcaaaaadaqadaqaauaa beqaceaaaeaacaWGRbGaeyOeI0YaaeWaaeaacqaHYoGydaqadaqaai aadMgacqGHRaWkcaaIXaaacaGLOaGaayzkaaGaeyOeI0IaaGymaaGa ayjkaiaawMcaaaqaaiaadUgaaaaacaGLOaGaayzkaaaaleaacaWGQb Gaeyypa0JaaGimaaqaaiaadUgaa0GaeyyeIuoaaSqaaiaadUgacaGG SaGaamiBaiabg2da9iaaicdaaeaacqGHEisPa0GaeyyeIuoakmaabm aabaqbaeqabiqaaaqaaiaadYgacqGHsislcaaIXaaabaGaamyBaaaa aiaawIcacaGLPaaadaqadaqaauaabeqaceaaaeaacaWGTbaabaGaam iCaaaaaiaawIcacaGLPaaadaqadaqaamaalaaabaGaeqiUdehabaGa eqiUdeNaey4kaSIaaGymaaaaaiaawIcacaGLPaaadaahaaWcbeqaai aadchaaaGcdaqadaqaauaabeqaceaaaeaacaWGRbaabaGaamOAaaaa aiaawIcacaGLPaaadaqadaqaauaabeqaceaaaeaadaqadaqaaiabek 7aInaabmaabaGaamyAaiabgUcaRiaaigdaaiaawIcacaGLPaaacqGH sislcaaIXaaacaGLOaGaayzkaaGaey4kaSIaam4AaaqaaiaadYgaaa aacaGLOaGaayzkaaGaamiuamaaBaaaleaacaWGQbGaaiilaiaadUga aeqaaOWaaeWaaeaacaaIXaGaey4kaSIaamiEaaGaayjkaiaawMcaai aadIhadaahaaWcbeqaaiaadchaaaGcdaqfGaqabSqabeaacqGHsisl cqaH4oqCcaWG4bWaaeWaaeaacaaIXaGaey4kaSIaamyBaaGaayjkai aawMcaaaqdbaGaamyzaaaaaSqaaiaadMgacqGH9aqpcaaIWaaabaGa eyOhIukaniabggHiLdaaleaacaWGTbGaeyypa0JaaGimaaqaaiabg6 HiLcqdcqGHris5aaWcbaGaamiCaiabg2da9iaaicdaaeaacqGHEisP a0GaeyyeIuoaaaa@CCCB@

This implies that:

f(x)= η i,j,k,l,m,p ( 1+x ) x p e θx( 1+m ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacaWG4bGaaiykaiabg2da9iabeE7aOnaaBaaaleaacaWGPbGaaiil aiaadQgacaGGSaGaam4AaiaacYcacaWGSbGaaiilaiaad2gacaGGSa GaamiCaaqabaGcdaqadaqaaiaaigdacqGHRaWkcaWG4baacaGLOaGa ayzkaaGaamiEamaaCaaaleqabaGaamiCaaaakmaavacabeWcbeqaai abgkHiTiabeI7aXjaadIhadaqadaqaaiaaigdacqGHRaWkcaWGTbaa caGLOaGaayzkaaaaneaacaWGLbaaaaaa@55F9@  (2.2.14)

Where

η i,j,k,l,m,p = αβ θ 2 ( θ+1 ) p=0 m=0 i=0 k,l=0 j=0 k ( 1 ) i+j+k+l+m α i+1 ( β( i+1 ) ) i!( β( i+1 )1j ) ( k( β( i+1 )1 ) k ) ( l1 m )( m p ) ( θ θ+1 ) p ( k j )( ( β( i+1 )1 )+k l ) P j,k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaeq4TdG2aaS baaSqaaiaadMgacaGGSaGaamOAaiaacYcacaWGRbGaaiilaiaadYga caGGSaGaamyBaiaacYcacaWGWbaabeaakiabg2da9maalaaabaGaeq ySdeMaeqOSdiMaeqiUde3aaWbaaSqabeaacaaIYaaaaaGcbaWaaeWa aeaacqaH4oqCcqGHRaWkcaaIXaaacaGLOaGaayzkaaaaamaaqahaba WaaabCaeaadaaeWbqaamaaqahabaWaaabCaeaadaWcaaqaamaabmaa baGaeyOeI0IaaGymaaGaayjkaiaawMcaamaaCaaaleqabaGaamyAai abgUcaRiaadQgacqGHRaWkcaWGRbGaey4kaSIaamiBaiabgUcaRiaa d2gaaaGccqaHXoqydaahaaWcbeqaaiaadMgacqGHRaWkcaaIXaaaaO WaaeWaaeaacqaHYoGydaqadaqaaiaadMgacqGHRaWkcaaIXaaacaGL OaGaayzkaaaacaGLOaGaayzkaaaabaGaamyAaiaacgcadaqadaqaai abek7aInaabmaabaGaamyAaiabgUcaRiaaigdaaiaawIcacaGLPaaa cqGHsislcaaIXaGaeyOeI0IaamOAaaGaayjkaiaawMcaaaaadaqada qaauaabeqaceaaaeaacaWGRbGaeyOeI0YaaeWaaeaacqaHYoGydaqa daqaaiaadMgacqGHRaWkcaaIXaaacaGLOaGaayzkaaGaeyOeI0IaaG ymaaGaayjkaiaawMcaaaqaaiaadUgaaaaacaGLOaGaayzkaaaaleaa caWGQbGaeyypa0JaaGimaaqaaiaadUgaa0GaeyyeIuoaaSqaaiaadU gacaGGSaGaamiBaiabg2da9iaaicdaaeaacqGHEisPa0GaeyyeIuoa kmaabmaabaqbaeqabiqaaaqaaiaadYgacqGHsislcaaIXaaabaGaam yBaaaaaiaawIcacaGLPaaadaqadaqaauaabeqaceaaaeaacaWGTbaa baGaamiCaaaaaiaawIcacaGLPaaadaqadaqaamaalaaabaGaeqiUde habaGaeqiUdeNaey4kaSIaaGymaaaaaiaawIcacaGLPaaadaahaaWc beqaaiaadchaaaGcdaqadaqaauaabeqaceaaaeaacaWGRbaabaGaam OAaaaaaiaawIcacaGLPaaadaqadaqaauaabeqaceaaaeaadaqadaqa aiabek7aInaabmaabaGaamyAaiabgUcaRiaaigdaaiaawIcacaGLPa aacqGHsislcaaIXaaacaGLOaGaayzkaaGaey4kaSIaam4Aaaqaaiaa dYgaaaaacaGLOaGaayzkaaGaamiuamaaBaaaleaacaWGQbGaaiilai aadUgaaeqaaaqaaiaadMgacqGH9aqpcaaIWaaabaGaeyOhIukaniab ggHiLdaaleaacaWGTbGaeyypa0JaaGimaaqaaiabg6HiLcqdcqGHri s5aaWcbaGaamiCaiabg2da9iaaicdaaeaacqGHEisPa0GaeyyeIuoa aaa@C4F8@

Hence,

μ n ' =E[ X n ]= 0 x n f(x)dx = 0 η i,j,k,l,m,p ( 1+x ) x n+p e θx( 1+m ) dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaWaaubmaeqale aacaWGUbaabaGaai4jaaqdbaGaeqiVd0gaaOGaeyypa0Jaamyramaa dmaabaWaaubiaeqaleqabaGaamOBaaqdbaGaamiwaaaaaOGaay5wai aaw2faaiabg2da9maapehabaWaaubiaeqaleqabaGaamOBaaqdbaGa amiEaaaakiaadAgacaGGOaGaamiEaiaacMcacaWGKbGaamiEaaWcba GaaGimaaqaaiabg6HiLcqdcqGHRiI8aOGaeyypa0Zaa8qCaeaacqaH 3oaAdaWgaaWcbaGaamyAaiaacYcacaWGQbGaaiilaiaadUgacaGGSa GaamiBaiaacYcacaWGTbGaaiilaiaadchaaeqaaOWaaeWaaeaacaaI XaGaey4kaSIaamiEaaGaayjkaiaawMcaaiaadIhadaahaaWcbeqaai aad6gacqGHRaWkcaWGWbaaaOWaaubiaeqaleqabaGaeyOeI0IaeqiU deNaamiEamaabmaabaGaaGymaiabgUcaRiaad2gaaiaawIcacaGLPa aaa0qaaiaadwgaaaGccaWGKbGaamiEaaWcbaGaaGimaaqaaiabg6Hi LcqdcqGHRiI8aaaa@71E1@

μ n ' =E[ X n ]= 0 x n f(x)dx = η i,j,k,l,m,p [ 0 x n+p e θx( 1+m ) dx+ 0 x n+p+1 e θx( 1+m ) dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaWaaubmaeqale aacaWGUbaabaGaai4jaaqdbaGaeqiVd0gaaOGaeyypa0Jaamyramaa dmaabaWaaubiaeqaleqabaGaamOBaaqdbaGaamiwaaaaaOGaay5wai aaw2faaiabg2da9maapehabaWaaubiaeqaleqabaGaamOBaaqdbaGa amiEaaaakiaadAgacaGGOaGaamiEaiaacMcacaWGKbGaamiEaaWcba GaaGimaaqaaiabg6HiLcqdcqGHRiI8aOGaeyypa0Jaeq4TdG2aaSba aSqaaiaadMgacaGGSaGaamOAaiaacYcacaWGRbGaaiilaiaadYgaca GGSaGaamyBaiaacYcacaWGWbaabeaakmaadmaabaWaa8qCaeaacaWG 4bWaaWbaaSqabeaacaWGUbGaey4kaSIaamiCaaaakiaadwgadaahaa WcbeqaaiabgkHiTiabeI7aXjaadIhadaqadaqaaiaaigdacqGHRaWk caWGTbaacaGLOaGaayzkaaaaaOGaamizaiaadIhacqGHRaWkdaWdXb qaaiaadIhadaahaaWcbeqaaiaad6gacqGHRaWkcaWGWbGaey4kaSIa aGymaaaakiaadwgadaahaaWcbeqaaiabgkHiTiabeI7aXjaadIhada qadaqaaiaaigdacqGHRaWkcaWGTbaacaGLOaGaayzkaaaaaOGaamiz aiaadIhaaSqaaiaaicdaaeaacqGHEisPa0Gaey4kIipaaSqaaiaaic daaeaacqGHEisPa0Gaey4kIipaaOGaay5waiaaw2faaaaa@8561@  (2.2.15)

Also, using integration by substitution method in equation (2.2.15); we obtain the following:

 Let u=θx( 1+m )x= u θ( 1+m ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamyDaiabg2 da9iabeI7aXjaadIhadaqadaqaaiaaigdacqGHRaWkcaWGTbaacaGL OaGaayzkaaGaeyO0H4TaamiEaiabg2da9maalaaabaGaamyDaaqaai abeI7aXnaabmaabaGaaGymaiabgUcaRiaad2gaaiaawIcacaGLPaaa aaaaaa@4B7A@ ; du dx =θ( 1+m ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGKbGaamyDaaqaaiaadsgacaWG4baaaiabg2da9iabeI7aXnaabmaa baGaaGymaiabgUcaRiaad2gaaiaawIcacaGLPaaaaaa@4224@  and dx= du θ( 1+m ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamizaiaadI hacqGH9aqpdaWcaaqaaiaadsgacaWG1baabaGaeqiUde3aaeWaaeaa caaIXaGaey4kaSIaamyBaaGaayjkaiaawMcaaaaaaaa@4224@

Substituting for u MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamyDaaaa@3871@ , x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@3874@ and dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamizaiaadI haaaa@395D@  in equation (2.2.15) and simplifying; we have:

μ n ' = η i,j,k,l,m,p [ 0 ( u θ( 1+m ) ) n+p e u ( du θ( 1+m ) )+ 0 ( u θ( 1+m ) ) n+p+1 e u ( du θ( 1+m ) ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaWaaubmaeqale aacaWGUbaabaGaai4jaaqdbaGaeqiVd0gaaOGaeyypa0Jaeq4TdG2a aSbaaSqaaiaadMgacaGGSaGaamOAaiaacYcacaWGRbGaaiilaiaadY gacaGGSaGaamyBaiaacYcacaWGWbaabeaakmaadmaabaWaa8qCaeaa daqadaqaamaalaaabaGaamyDaaqaaiabeI7aXnaabmaabaGaaGymai abgUcaRiaad2gaaiaawIcacaGLPaaaaaaacaGLOaGaayzkaaWaaWba aSqabeaacaWGUbGaey4kaSIaamiCaaaakiaadwgadaahaaWcbeqaai abgkHiTiaadwhaaaGcdaqadaqaamaalaaabaGaamizaiaadwhaaeaa cqaH4oqCdaqadaqaaiaaigdacqGHRaWkcaWGTbaacaGLOaGaayzkaa aaaaGaayjkaiaawMcaaiabgUcaRmaapehabaWaaeWaaeaadaWcaaqa aiaadwhaaeaacqaH4oqCdaqadaqaaiaaigdacqGHRaWkcaWGTbaaca GLOaGaayzkaaaaaaGaayjkaiaawMcaamaaCaaaleqabaGaamOBaiab gUcaRiaadchacqGHRaWkcaaIXaaaaOGaamyzamaaCaaaleqabaGaey OeI0IaamyDaaaakmaabmaabaWaaSaaaeaacaWGKbGaamyDaaqaaiab eI7aXnaabmaabaGaaGymaiabgUcaRiaad2gaaiaawIcacaGLPaaaaa aacaGLOaGaayzkaaaaleaacaaIWaaabaGaeyOhIukaniabgUIiYdaa leaacaaIWaaabaGaeyOhIukaniabgUIiYdaakiaawUfacaGLDbaaaa a@8442@

μ n ' = η i,j,k,l,m,p [ 1 ( θ( 1+m ) ) n+p+1 0 u n+p e u du+ 1 ( θ( 1+m ) ) n+p+2 0 u n+p+1 e u du ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaWaaubmaeqale aacaWGUbaabaGaai4jaaqdbaGaeqiVd0gaaOGaeyypa0Jaeq4TdG2a aSbaaSqaaiaadMgacaGGSaGaamOAaiaacYcacaWGRbGaaiilaiaadY gacaGGSaGaamyBaiaacYcacaWGWbaabeaakmaadmaabaWaaSaaaeaa caaIXaaabaWaaeWaaeaacqaH4oqCdaqadaqaaiaaigdacqGHRaWkca WGTbaacaGLOaGaayzkaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaWG UbGaey4kaSIaamiCaiabgUcaRiaaigdaaaaaaOWaa8qCaeaacaWG1b WaaWbaaSqabeaacaWGUbGaey4kaSIaamiCaaaakiaadwgadaahaaWc beqaaiabgkHiTiaadwhaaaGccaWGKbGaamyDaiabgUcaRmaalaaaba GaaGymaaqaamaabmaabaGaeqiUde3aaeWaaeaacaaIXaGaey4kaSIa amyBaaGaayjkaiaawMcaaaGaayjkaiaawMcaamaaCaaaleqabaGaam OBaiabgUcaRiaadchacqGHRaWkcaaIYaaaaaaakmaapehabaGaamyD amaaCaaaleqabaGaamOBaiabgUcaRiaadchacqGHRaWkcaaIXaaaaO GaamyzamaaCaaaleqabaGaeyOeI0IaamyDaaaakiaadsgacaWG1baa leaacaaIWaaabaGaeyOhIukaniabgUIiYdaaleaacaaIWaaabaGaey OhIukaniabgUIiYdaakiaawUfacaGLDbaaaaa@8027@

μ n ' = η i,j,k,l,m,p [ 1 ( θ( 1+m ) ) n+p+1 0 u n+p+11 e u du+ 1 ( θ( 1+m ) ) n+p+2 0 u n+p+21 e u du ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaWaaubmaeqale aacaWGUbaabaGaai4jaaqdbaGaeqiVd0gaaOGaeyypa0Jaeq4TdG2a aSbaaSqaaiaadMgacaGGSaGaamOAaiaacYcacaWGRbGaaiilaiaadY gacaGGSaGaamyBaiaacYcacaWGWbaabeaakmaadmaabaWaaSaaaeaa caaIXaaabaWaaeWaaeaacqaH4oqCdaqadaqaaiaaigdacqGHRaWkca WGTbaacaGLOaGaayzkaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaWG UbGaey4kaSIaamiCaiabgUcaRiaaigdaaaaaaOWaa8qCaeaacaWG1b WaaWbaaSqabeaacaWGUbGaey4kaSIaamiCaiabgUcaRiaaigdacqGH sislcaaIXaaaaOGaamyzamaaCaaaleqabaGaeyOeI0IaamyDaaaaki aadsgacaWG1bGaey4kaSYaaSaaaeaacaaIXaaabaWaaeWaaeaacqaH 4oqCdaqadaqaaiaaigdacqGHRaWkcaWGTbaacaGLOaGaayzkaaaaca GLOaGaayzkaaWaaWbaaSqabeaacaWGUbGaey4kaSIaamiCaiabgUca RiaaikdaaaaaaOWaa8qCaeaacaWG1bWaaWbaaSqabeaacaWGUbGaey 4kaSIaamiCaiabgUcaRiaaikdacqGHsislcaaIXaaaaOGaamyzamaa CaaaleqabaGaeyOeI0IaamyDaaaakiaadsgacaWG1baaleaacaaIWa aabaGaeyOhIukaniabgUIiYdaaleaacaaIWaaabaGaeyOhIukaniab gUIiYdaakiaawUfacaGLDbaaaaa@8515@  (2.2.16)

Again recall that 0 t k1 e t dt=Γ( k ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaaca WG0bWaaWbaaSqabeaacaWGRbGaeyOeI0IaaGymaaaakiaadwgadaah aaWcbeqaaiabgkHiTiaadshaaaGccaWGKbGaamiDaiabg2da9iabfo 5ahnaabmaabaGaam4AaaGaayjkaiaawMcaaaWcbaGaaGimaaqaaiab g6HiLcqdcqGHRiI8aaaa@49A3@ and that 0 t k e t dt= 0 t k+11 e t dt=Γ( k+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaaca WG0bWaaWbaaSqabeaacaWGRbaaaOGaamyzamaaCaaaleqabaGaeyOe I0IaamiDaaaakiaadsgacaWG0bGaeyypa0Zaa8qCaeaacaWG0bWaaW baaSqabeaacaWGRbGaey4kaSIaaGymaiabgkHiTiaaigdaaaGccaWG LbWaaWbaaSqabeaacqGHsislcaWG0baaaOGaamizaiaadshacqGH9a qpcqqHtoWrdaqadaqaaiaadUgacqGHRaWkcaaIXaaacaGLOaGaayzk aaaaleaacaaIWaaabaGaeyOhIukaniabgUIiYdaaleaacaaIWaaaba GaeyOhIukaniabgUIiYdaaaa@5980@

Thus we obtain the nth ordinary moment of X for the Weibull-Lindley distribution as follows:

μ n ' = η i,j,k,l,m,p [ Γ( n+p+1 ) ( θ( 1+m ) ) n+p+1 + Γ( n+p+2 ) ( θ( 1+m ) ) n+p+2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaWaaubmaeqale aacaWGUbaabaGaai4jaaqdbaGaeqiVd0gaaOGaeyypa0Jaeq4TdG2a aSbaaSqaaiaadMgacaGGSaGaamOAaiaacYcacaWGRbGaaiilaiaadY gacaGGSaGaamyBaiaacYcacaWGWbaabeaakmaadmaabaWaaSaaaeaa cqqHtoWrdaqadaqaaiaad6gacqGHRaWkcaWGWbGaey4kaSIaaGymaa GaayjkaiaawMcaaaqaamaabmaabaGaeqiUde3aaeWaaeaacaaIXaGa ey4kaSIaamyBaaGaayjkaiaawMcaaaGaayjkaiaawMcaamaaCaaale qabaGaamOBaiabgUcaRiaadchacqGHRaWkcaaIXaaaaaaakiabgUca RmaalaaabaGaeu4KdC0aaeWaaeaacaWGUbGaey4kaSIaamiCaiabgU caRiaaikdaaiaawIcacaGLPaaaaeaadaqadaqaaiabeI7aXnaabmaa baGaaGymaiabgUcaRiaad2gaaiaawIcacaGLPaaaaiaawIcacaGLPa aadaahaaWcbeqaaiaad6gacqGHRaWkcaWGWbGaey4kaSIaaGOmaaaa aaaakiaawUfacaGLDbaaaaa@70C7@  (2.2.17)

The Mean

The mean of the WLnD can be obtained from the nth moment of the distribution when n=1 as follows:

μ n ' = η i,j,k,l,m,p [ Γ( n+p+1 ) ( θ( 1+m ) ) n+p+1 + Γ( n+p+2 ) ( θ( 1+m ) ) n+p+2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaWaaubmaeqale aacaWGUbaabaGaai4jaaqdbaGaeqiVd0gaaOGaeyypa0Jaeq4TdG2a aSbaaSqaaiaadMgacaGGSaGaamOAaiaacYcacaWGRbGaaiilaiaadY gacaGGSaGaamyBaiaacYcacaWGWbaabeaakmaadmaabaWaaSaaaeaa cqqHtoWrdaqadaqaaiaad6gacqGHRaWkcaWGWbGaey4kaSIaaGymaa GaayjkaiaawMcaaaqaamaabmaabaGaeqiUde3aaeWaaeaacaaIXaGa ey4kaSIaamyBaaGaayjkaiaawMcaaaGaayjkaiaawMcaamaaCaaale qabaGaamOBaiabgUcaRiaadchacqGHRaWkcaaIXaaaaaaakiabgUca RmaalaaabaGaeu4KdC0aaeWaaeaacaWGUbGaey4kaSIaamiCaiabgU caRiaaikdaaiaawIcacaGLPaaaaeaadaqadaqaaiabeI7aXnaabmaa baGaaGymaiabgUcaRiaad2gaaiaawIcacaGLPaaaaiaawIcacaGLPa aadaahaaWcbeqaaiaad6gacqGHRaWkcaWGWbGaey4kaSIaaGOmaaaa aaaakiaawUfacaGLDbaaaaa@70C7@

E[ X ]= μ 1 ' = η i,j,k,l,m,p [ Γ( p+2 ) ( θ( 1+m ) ) p+2 + Γ( p+3 ) ( θ( 1+m ) ) p+3 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamyramaadm aabaGaamiwaaGaay5waiaaw2faaiabg2da9maavadabeWcbaGaaGym aaqaaiaacEcaa0qaaiabeY7aTbaakiabg2da9iabeE7aOnaaBaaale aacaWGPbGaaiilaiaadQgacaGGSaGaam4AaiaacYcacaWGSbGaaiil aiaad2gacaGGSaGaamiCaaqabaGcdaWadaqaamaalaaabaGaeu4KdC 0aaeWaaeaacaWGWbGaey4kaSIaaGOmaaGaayjkaiaawMcaaaqaamaa bmaabaGaeqiUde3aaeWaaeaacaaIXaGaey4kaSIaamyBaaGaayjkai aawMcaaaGaayjkaiaawMcaamaaCaaaleqabaGaamiCaiabgUcaRiaa ikdaaaaaaOGaey4kaSYaaSaaaeaacqqHtoWrdaqadaqaaiaadchacq GHRaWkcaaIZaaacaGLOaGaayzkaaaabaWaaeWaaeaacqaH4oqCdaqa daqaaiaaigdacqGHRaWkcaWGTbaacaGLOaGaayzkaaaacaGLOaGaay zkaaWaaWbaaSqabeaacaWGWbGaey4kaSIaaG4maaaaaaaakiaawUfa caGLDbaaaaa@6DDE@  (2.2.17)

The Variance

The nth central moment or moment about the mean of X, say μ n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaGqacabaaaaaaaaapeGaa8NBaaWdaeqaaaaa@3A83@ , can be obtained as

μ n =E [ X μ 1 ' ] n = i=0 n (1) i ( n i ) μ 1 'i μ ni ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaWaaubeaeqale aacaWGUbaabeqdbaGaeqiVd0gaaOGaeyypa0JaamyramaadmaabaGa amiwaiabgkHiTiabeY7aTnaaDaaaleaacaaIXaaabaGaai4jaaaaaO Gaay5waiaaw2faamaaCaaaleqabaGaamOBaaaakiabg2da9maaqaha baGaaiikaiabgkHiTiaaigdacaGGPaWaaWbaaSqabeaacaWGPbaaaO WaaeWaaeaafaqabeGabaaabaGaamOBaaqaaiaadMgaaaaacaGLOaGa ayzkaaGaeqiVd02aa0baaSqaaiaaigdaaeaacaGGNaGaamyAaaaaki abeY7aTnaaDaaaleaacaWGUbGaeyOeI0IaamyAaaqaaiaacEcaaaaa baGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdaaaa@5C91@  (2.2.18)

The variance of X for WLnD is obtained from the nth central moment when n=2, that is, the variance of X is the nth central moment of order two (n=2) and is given as follows:

Var(X)=E[ X 2 ] { E[X] } 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kucpgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamOvaiaadg gacaWGYbGaaiikaiaadIfacaGGPaGaeyypa0JaamyraiaacUfacaWG ybWaaWbaaSqabeaacaaIYaaaaOGaaiyxaiabgkHiTmaacmaabaGaam yraiaacUfacaWGybGaaiyxaaGaay5Eaiaaw2haamaaCaaaleqabaGa aGOmaaaaaaa@4933@ (2.2.19)

Var(X)= μ 2 ' { μ 1 ' } 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamOvaiaadg gacaWGYbGaaiikaiaadIfacaGGPaGaeyypa0JaeqiVd02aaSbaaSqa aiaaikdaaeqaaOWaaWbaaSqabeaacaGGNaaaaOGaeyOeI0YaaiWaae aacqaH8oqBdaWgaaWcbaGaaGymaaqabaGcdaahaaWcbeqaaiaacEca aaaakiaawUhacaGL9baadaahaaWcbeqaaiaaikdaaaaaaa@4885@

Var(X)= η i,j,k,l,m,p [ Γ( p+3 ) ( θ( 1+m ) ) p+3 + Γ( p+4 ) ( θ( 1+m ) ) p+4 ] { η i,j,k,l,m,p [ Γ( p+2 ) ( θ( 1+m ) ) p+2 + Γ( p+3 ) ( θ( 1+m ) ) p+3 ] } 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamOvaiaadg gacaWGYbGaaiikaiaadIfacaGGPaGaeyypa0Jaeq4TdG2aaSbaaSqa aiaadMgacaGGSaGaamOAaiaacYcacaWGRbGaaiilaiaadYgacaGGSa GaamyBaiaacYcacaWGWbaabeaakmaadmaabaWaaSaaaeaacqqHtoWr daqadaqaaiaadchacqGHRaWkcaaIZaaacaGLOaGaayzkaaaabaWaae WaaeaacqaH4oqCdaqadaqaaiaaigdacqGHRaWkcaWGTbaacaGLOaGa ayzkaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaWGWbGaey4kaSIaaG 4maaaaaaGccqGHRaWkdaWcaaqaaiabfo5ahnaabmaabaGaamiCaiab gUcaRiaaisdaaiaawIcacaGLPaaaaeaadaqadaqaaiabeI7aXnaabm aabaGaaGymaiabgUcaRiaad2gaaiaawIcacaGLPaaaaiaawIcacaGL PaaadaahaaWcbeqaaiaadchacqGHRaWkcaaI0aaaaaaaaOGaay5wai aaw2faaiabgkHiTmaacmaabaGaeq4TdG2aaSbaaSqaaiaadMgacaGG SaGaamOAaiaacYcacaWGRbGaaiilaiaadYgacaGGSaGaamyBaiaacY cacaWGWbaabeaakmaadmaabaWaaSaaaeaacqqHtoWrdaqadaqaaiaa dchacqGHRaWkcaaIYaaacaGLOaGaayzkaaaabaWaaeWaaeaacqaH4o qCdaqadaqaaiaaigdacqGHRaWkcaWGTbaacaGLOaGaayzkaaaacaGL OaGaayzkaaWaaWbaaSqabeaacaWGWbGaey4kaSIaaGOmaaaaaaGccq GHRaWkdaWcaaqaaiabfo5ahnaabmaabaGaamiCaiabgUcaRiaaioda aiaawIcacaGLPaaaaeaadaqadaqaaiabeI7aXnaabmaabaGaaGymai abgUcaRiaad2gaaiaawIcacaGLPaaaaiaawIcacaGLPaaadaahaaWc beqaaiaadchacqGHRaWkcaaIZaaaaaaaaOGaay5waiaaw2faaaGaay 5Eaiaaw2haamaaCaaaleqabaGaaGOmaaaaaaa@9BE4@  (2.2.20)

The coefficients variation, skewness and kurtosis measures can also be calculated from the non-central moments using some well-known relationships.

Moment generating function

The mgf of a random variable X can be obtained by

M x (t)=E[ e tx ]= 0 e tx f(x)dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaWaaubeaeqale aacaWG4baabeqdbaGaamytaaaakiaacIcacaWG0bGaaiykaiabg2da 9iaadweadaWadaqaamaavacabeWcbeqaaiaadshacaWG4baaneaaca WGLbaaaaGccaGLBbGaayzxaaGaeyypa0Zaa8qCaeaadaqfGaqabSqa beaacaWG0bGaamiEaaqdbaGaamyzaaaakiaadAgacaGGOaGaamiEai aacMcacaWGKbGaamiEaaWcbaGaaGimaaqaaiabg6HiLcqdcqGHRiI8 aaaa@50EB@  (2.2.21)

Using power series expansion in equation (2.2.21) and simplifying the integral therefore we have;

M x (t)= n=0 t n n! μ n ' = n=0 t n n! { η i,j,k,l,m,p [ Γ( n+p+1 ) ( θ( 1+m ) ) n+p+1 + Γ( n+p+2 ) ( θ( 1+m ) ) n+p+2 ] } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWG4baabeaakiaacIcacaWG0bGaaiykaiabg2da9maaqaha baWaaSaaaeaacaWG0bWaaWbaaSqabeaacaWGUbaaaaGcbaGaamOBai aacgcaaaGaeqiVd02aa0baaSqaaiaad6gaaeaacaGGNaaaaaqaaiaa d6gacqGH9aqpcaaIWaaabaGaeyOhIukaniabggHiLdGccqGH9aqpda aeWbqaamaalaaabaGaamiDamaaCaaaleqabaGaamOBaaaaaOqaaiaa d6gacaGGHaaaamaacmaabaGaeq4TdG2aaSbaaSqaaiaadMgacaGGSa GaamOAaiaacYcacaWGRbGaaiilaiaadYgacaGGSaGaamyBaiaacYca caWGWbaabeaakmaadmaabaWaaSaaaeaacqqHtoWrdaqadaqaaiaad6 gacqGHRaWkcaWGWbGaey4kaSIaaGymaaGaayjkaiaawMcaaaqaamaa bmaabaGaeqiUde3aaeWaaeaacaaIXaGaey4kaSIaamyBaaGaayjkai aawMcaaaGaayjkaiaawMcaamaaCaaaleqabaGaamOBaiabgUcaRiaa dchacqGHRaWkcaaIXaaaaaaakiabgUcaRmaalaaabaGaeu4KdC0aae WaaeaacaWGUbGaey4kaSIaamiCaiabgUcaRiaaikdaaiaawIcacaGL PaaaaeaadaqadaqaaiabeI7aXnaabmaabaGaaGymaiabgUcaRiaad2 gaaiaawIcacaGLPaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaad6ga cqGHRaWkcaWGWbGaey4kaSIaaGOmaaaaaaaakiaawUfacaGLDbaaai aawUhacaGL9baaaSqaaiaad6gacqGH9aqpcaaIWaaabaGaeyOhIuka niabggHiLdaaaa@8C74@  (2.2.22)

where n and t are constants, t is a real number and μ n ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaacbiaeaaaaaa aaa8qacaWF8oWdamaaDaaaleaapeGaa8NBaaWdaeaapeGaa83jaaaa aaa@3AE4@  denotes the nth ordinary moment of X .

Characteristics function

The characteristics function of a random variable X is given by;

φ x (t)=E[ e itx ]=E[ cos(tx)+isin(tx) ]=E[ cos(tx) ]+E[ isin(tx) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaeqOXdO2aaS baaSqaaiaadIhaaeqaaOGaaiikaiaadshacaGGPaGaeyypa0Jaamyr amaadmaabaGaamyzamaaCaaaleqabaGaamyAaiaadshacaWG4baaaa GccaGLBbGaayzxaaGaeyypa0JaamyramaadmaabaGaci4yaiaac+ga caGGZbGaaiikaiaadshacaWG4bGaaiykaiabgUcaRiaadMgaciGGZb GaaiyAaiaac6gacaGGOaGaamiDaiaadIhacaGGPaaacaGLBbGaayzx aaGaeyypa0JaamyramaadmaabaGaci4yaiaac+gacaGGZbGaaiikai aadshacaWG4bGaaiykaaGaay5waiaaw2faaiabgUcaRiaadweadaWa daqaaiaadMgaciGGZbGaaiyAaiaac6gacaGGOaGaamiDaiaadIhaca GGPaaacaGLBbGaayzxaaaaaa@6AF2@  (2.2.23)

Simple algebra and power series expansion proves that

ϕ x (t)= n=0 ( 1 ) n t 2n ( 2n )! μ 2n ' +i n=0 ( 1 ) n t 2n+1 ( 2n+1 )! μ 2n+1 ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaWaaubeaeqale aacaWG4baabeqdbaGaeqy1dygaaOGaaiikaiaadshacaGGPaGaeyyp a0ZaaabCaeaadaWcaaqaamaabmaabaGaeyOeI0IaaGymaaGaayjkai aawMcaamaaCaaaleqabaGaamOBaaaakiaadshadaahaaWcbeqaaiaa ikdacaWGUbaaaaGcbaWaaeWaaeaacaaIYaGaamOBaaGaayjkaiaawM caaiaacgcaaaWaaubmaeqaleaacaaIYaGaamOBaaqaaiaacEcaa0qa aiabeY7aTbaaaSqaaiaad6gacqGH9aqpcaaIWaaabaGaeyOhIukani abggHiLdGccqGHRaWkcaWGPbWaaabCaeaadaWcaaqaamaabmaabaGa eyOeI0IaaGymaaGaayjkaiaawMcaamaaCaaaleqabaGaamOBaaaaki aadshadaahaaWcbeqaaiaaikdacaWGUbGaey4kaSIaaGymaaaaaOqa amaabmaabaGaaGOmaiaad6gacqGHRaWkcaaIXaaacaGLOaGaayzkaa GaaiyiaaaadaqfWaqabSqaaiaaikdacaWGUbGaey4kaSIaaGymaaqa aiaacEcaa0qaaiabeY7aTbaaaSqaaiaad6gacqGH9aqpcaaIWaaaba GaeyOhIukaniabggHiLdaaaa@7099@  (2.2.24)

Where μ 2n ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaacbiaeaaaaaa aaa8qacaWF8oWdamaaDaaaleaapeGaaGOmaiaa=5gaa8aabaWdbiaa =Dcaaaaaaa@3BA0@  and μ 2n+1 ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaacbiaeaaaaaa aaa8qacaWF8oWdamaaDaaaleaapeGaaGOmaiaa=5gacqGHRaWkcaaI Xaaapaqaa8qacaWFNaaaaaaa@3D3D@ are the moments of X for n=2n and n=2n+1 respectively and can be obtained from μ n ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaacbiaeaaaaaa aaa8qacaWF8oWdamaaDaaaleaapeGaa8NBaaWdaeaapeGaa83jaaaa aaa@3AE4@  in equation (2.2.17)

Some reliability functions

In this section, we present some reliability functions associated with WLnD including the survival and hazard functions.

The Survival function

The survival function describes the likelihood that a system or an individual will not fail after a given time. It tells us about the probability of success or survival of a given product or component. Mathematically, the survival function is given by:

S( x )=1F( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaqcaaKaam4uaO WaaeWaaKaaafaacaWG4baacaGLOaGaayzkaaGaeyypa0JaaGymaiab gkHiTiaadAeakmaabmaajaaqbaGaamiEaaGaayjkaiaawMcaaaaa@41C3@  (2.3.1)

Taking F(x) to be the cdf of the Weibull-Lindley distribution, substituting and simplifying (2.3.1) above, we get the survival function of the WLnD as:

S( x )=1 e α { log[ 1[ 1+ θx θ+1 ] e θx ] } β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaqcKbaG=laado fakmaabmaajqgaa+FaaiaadIhaaiaawIcacaGLPaaacqGH9aqpcaaI XaGaeyOeI0IcdaqfGaqabSqabeaacqGHsislcqaHXoqydaGadaqaai abgkHiTiGacYgacaGGVbGaai4zamaadmaabaGaaGymaiabgkHiTmaa dmaabaGaaGymaiabgUcaRmaalaaabaGaeqiUdeNaamiEaaqaaiabeI 7aXjabgUcaRiaaigdaaaaacaGLBbGaayzxaaWaaubiaeqameqabaGa eyOeI0IaeqiUdeNaamiEaaGdbaGaamyzaaaaaSGaay5waiaaw2faaa Gaay5Eaiaaw2haamaaCaaameqabaGaeqOSdigaaaqdbaGaamyzaaaa aaa@5F24@  (2.3.2)

Below is a plot of the survival function at chosen parameter values in Figure 3. The figure above revealed that the probability of survival for any random variable following a Weibull-Lindley distribution decreases as the values of the random variable increases, that is, as time goes on, probability of life decreases. This implies that the Weibull-Lindley distribution can be used to model random variables whose survival rate decreases as their age grows.

Figure 3 Plot of the survival function of the WLnD for different parameter values.

The Hazard function

Hazard function as the name implies is also called risk function, it gives us the probability that a component will fail or die for an interval of time. The hazard function is defined mathematically as;

h( x )= f( x ) 1F( x ) = f( x ) S( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamiAamaabm aabaGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaabaGaamOzamaa bmaabaGaamiEaaGaayjkaiaawMcaaaqaaiaaigdacqGHsislcaWGgb WaaeWaaeaacaWG4baacaGLOaGaayzkaaaaaiabg2da9maalaaabaGa amOzamaabmaabaGaamiEaaGaayjkaiaawMcaaaqaaiaadofadaqada qaaiaadIhaaiaawIcacaGLPaaaaaaaaa@4C4F@  (2.3.3)

Taking f(x) and F(x) to be the pdf and cdf of the proposed Weibull-Lindley distribution given previously, we obtain the hazard function as:

h( x )= αβ θ 2 ( 1+x ) e θx { log[ 1[ 1+ θx θ+1 ] e θx ] } β1 e α { log[ 1[ 1+ θx θ+1 ] e θx ] } β ( θ+1 )[ 1[ 1+ θx θ+1 ] e θx ]( 1 e α { log[ 1[ 1+ θx θ+1 ] e θx ] } β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamiAamaabm aabaGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqySdeMa eqOSdiMaeqiUde3aaWbaaSqabeaacaaIYaaaaOWaaeWaaeaacaaIXa Gaey4kaSIaamiEaaGaayjkaiaawMcaamaavacabeWcbeqaaiabgkHi TiabeI7aXjaadIhaa0qaaiaadwgaaaGcdaGadaqaaiabgkHiTiGacY gacaGGVbGaai4zamaadmaabaGaaGymaiabgkHiTmaadmaabaGaaGym aiabgUcaRmaalaaabaGaeqiUdeNaamiEaaqaaiabeI7aXjabgUcaRi aaigdaaaaacaGLBbGaayzxaaWaaubiaeqaleqabaGaeyOeI0IaeqiU deNaamiEaaqdbaGaamyzaaaaaOGaay5waiaaw2faaaGaay5Eaiaaw2 haamaaCaaaleqabaGaeqOSdiMaeyOeI0IaaGymaaaakmaavacabeWc beqaaiabgkHiTiabeg7aHnaacmaabaGaeyOeI0IaciiBaiaac+gaca GGNbWaamWaaeaacaaIXaGaeyOeI0YaamWaaeaacaaIXaGaey4kaSYa aSaaaeaacqaH4oqCcaWG4baabaGaeqiUdeNaey4kaSIaaGymaaaaai aawUfacaGLDbaadaqfGaqabWqabeaacqGHsislcqaH4oqCcaWG4baa oeaacaWGLbaaaaWccaGLBbGaayzxaaaacaGL7bGaayzFaaWaaWbaaW qabeaacqaHYoGyaaaaneaacaWGLbaaaaGcbaWaaeWaaeaacqaH4oqC cqGHRaWkcaaIXaaacaGLOaGaayzkaaWaamWaaeaacaaIXaGaeyOeI0 YaamWaaeaacaaIXaGaey4kaSYaaSaaaeaacqaH4oqCcaWG4baabaGa eqiUdeNaey4kaSIaaGymaaaaaiaawUfacaGLDbaadaqfGaqabSqabe aacqGHsislcqaH4oqCcaWG4baaneaacaWGLbaaaaGccaGLBbGaayzx aaWaaeWaaeaacaaIXaGaeyOeI0YaaubiaeqaleqabaGaeyOeI0Iaeq ySde2aaiWaaeaacqGHsislciGGSbGaai4BaiaacEgadaWadaqaaiaa igdacqGHsisldaWadaqaaiaaigdacqGHRaWkdaWcaaqaaiabeI7aXj aadIhaaeaacqaH4oqCcqGHRaWkcaaIXaaaaaGaay5waiaaw2faamaa vacabeadbeqaaiabgkHiTiabeI7aXjaadIhaa4qaaiaadwgaaaaali aawUfacaGLDbaaaiaawUhacaGL9baadaahaaadbeqaaiabek7aIbaa a0qaaiaadwgaaaaakiaawIcacaGLPaaaaaaaaa@BC41@ (2.3.4)

The following is a plot of the hazard function at chosen parameter values in Figure 4.

Figure 4 Plot of the hazard function of the WLnD for different parameter values.

Figure 4 above shows the behavior of hazard function of the WLnD and it means that the probability of failure for any Weibull-Lindley random variable increases as the time or age of the variable grows or increases, that is, as time goes, the probability of failure or death increases and becomes constant after some times.

Order statistics

Order statistics are used widely over the years for solving a huge set of problems such as in robust statistical estimation and detection of outliers, characterization of probability distributions and goodness of fit tests, entropy estimation, analyses of censored samples, reliability analysis, quality control and strength of materials. Suppose X 1 , X 2 ,......, X n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaBa aaleaacaaIXaaabeaakiaacYcacaWGybWaaSbaaSqaaiaaikdaaeqa aOGaaiilaiaac6cacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaiaacY cacaWGybWaaSbaaSqaaiaad6gaaeqaaaaa@434C@  is a random sample from a distribution with pdf, f(x), and let X 1:n < X 2:n <......< X a:n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaBa aaleaacaaIXaGaaiOoaiaad6gaaeqaaOGaeyipaWJaamiwamaaBaaa leaacaaIYaGaaiOoaiaad6gaaeqaaOGaeyipaWJaaiOlaiaac6caca GGUaGaaiOlaiaac6cacaGGUaGaeyipaWJaamiwamaaBaaaleaacaWG HbGaaiOoaiaad6gaaeqaaaaa@494E@  denote the corresponding order statistic obtained from this sample. The pdf, f a:n ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaWGHbGaaiOoaiaad6gaaeqaaOWaaeWaaeaacaWG4baacaGL OaGaayzkaaaaaa@3DB5@  of the a th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaamiDaiaadIgaaaaaaa@3A70@  order statistic can be defined as;

f a:n ( x )= n! ( a1 )!( na )! k=0 na (1) k ( na k ) f( x ) F( x ) k+a1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaWaaubeaeqale aacaWGHbGaaiOoaiaad6gaaeqaneaacaWGMbaaaOWaaeWaaeaacaWG 4baacaGLOaGaayzkaaGaeyypa0ZaaSaaaeaacaWGUbGaaiyiaaqaam aabmaabaGaamyyaiabgkHiTiaaigdaaiaawIcacaGLPaaacaGGHaWa aeWaaeaacaWGUbGaeyOeI0IaamyyaaGaayjkaiaawMcaaiaacgcaaa WaaabCaeaadaqfGaqabSqabeaacaWGRbaaneaacaGGOaGaeyOeI0Ia aGymaiaacMcaaaGcdaqadaqaauaabeqaceaaaeaacaWGUbGaeyOeI0 IaamyyaaqaaiaadUgaaaaacaGLOaGaayzkaaaaleaacaWGRbGaeyyp a0JaaGimaaqaaiaad6gacqGHsislcaWGHbaaniabggHiLdGccaWGMb WaaeWaaeaacaWG4baacaGLOaGaayzkaaWaaubiaeqaleqabaGaam4A aiabgUcaRiaadggacqGHsislcaaIXaaaneaacaWGgbWaaeWaaeaaca WG4baacaGLOaGaayzkaaaaaaaa@6704@  (2.4.1)

where f(x) and F(x) are the pdf and cdf of the Weibull-Lindley distribution respectively.

Using (2.1.3) and (2.1.4), the pdf of the a th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaahaa WcbeqaaiaadshacaWGObaaaaaa@38E4@  order statistics X a:n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIfadaWgaa WcbaGaamyyaiaacQdacaWGUbaabeaaaaa@398B@ , can be expressed from (2.4.1) as;

f a:n (x)= n! (a1)!(na)! k=0 na (1) k ( na k ) [ αβ θ 2 ( 1+x ) e θx { log[ 1[ 1+ θx θ+1 ] e θx ] } β1 ( θ+1 )[ 1[ 1+ θx θ+1 ] e θx ] e α { log[ 1[ 1+ θx θ+1 ] e θx ] } β ]* [ e α { log[ 1[ 1+ θx θ+1 ] e θx ] } β ] a+k1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaWaaubeaeqale aacaWGHbGaaiOoaiaad6gaaeqaneaacaWGMbaaaOGaaiikaiaadIha caGGPaGaeyypa0ZaaSaaaeaacaWGUbGaaiyiaaqaaiaacIcacaWGHb GaeyOeI0IaaGymaiaacMcacaGGHaGaaiikaiaad6gacqGHsislcaWG HbGaaiykaiaacgcaaaWaaabCaeaadaqfGaqabSqabeaacaWGRbaane aacaGGOaGaeyOeI0IaaGymaiaacMcaaaGcdaqadaqaauaabeqaceaa aeaacaWGUbGaeyOeI0IaamyyaaqaaiaadUgaaaaacaGLOaGaayzkaa aaleaacaWGRbGaeyypa0JaaGimaaqaaiaad6gacqGHsislcaWGHbaa niabggHiLdGcdaWadaqaamaalaaabaGaeqySdeMaeqOSdiMaeqiUde 3aaWbaaSqabeaacaaIYaaaaOWaaeWaaeaacaaIXaGaey4kaSIaamiE aaGaayjkaiaawMcaamaavacabeWcbeqaaiabgkHiTiabeI7aXjaadI haa0qaaiaadwgaaaGcdaGadaqaaiabgkHiTiGacYgacaGGVbGaai4z amaadmaabaGaaGymaiabgkHiTmaadmaabaGaaGymaiabgUcaRmaala aabaGaeqiUdeNaamiEaaqaaiabeI7aXjabgUcaRiaaigdaaaaacaGL BbGaayzxaaWaaubiaeqaleqabaGaeyOeI0IaeqiUdeNaamiEaaqdba GaamyzaaaaaOGaay5waiaaw2faaaGaay5Eaiaaw2haamaaCaaaleqa baGaeqOSdiMaeyOeI0IaaGymaaaaaOqaamaabmaabaGaeqiUdeNaey 4kaSIaaGymaaGaayjkaiaawMcaamaadmaabaGaaGymaiabgkHiTmaa dmaabaGaaGymaiabgUcaRmaalaaabaGaeqiUdeNaamiEaaqaaiabeI 7aXjabgUcaRiaaigdaaaaacaGLBbGaayzxaaWaaubiaeqaleqabaGa eyOeI0IaeqiUdeNaamiEaaqdbaGaamyzaaaaaOGaay5waiaaw2faam aavacabeWcbeqaaiabeg7aHnaacmaabaGaeyOeI0IaciiBaiaac+ga caGGNbWaamWaaeaacaaIXaGaeyOeI0YaamWaaeaacaaIXaGaey4kaS YaaSaaaeaacqaH4oqCcaWG4baabaGaeqiUdeNaey4kaSIaaGymaaaa aiaawUfacaGLDbaadaqfGaqabWqabeaacqGHsislcqaH4oqCcaWG4b aaoeaacaWGLbaaaaWccaGLBbGaayzxaaaacaGL7bGaayzFaaWaaWba aWqabeaacqaHYoGyaaaaneaacaWGLbaaaaaaaOGaay5waiaaw2faai aacQcadaqfGaqabSqabeaacaWGHbGaey4kaSIaam4AaiabgkHiTiaa igdaa0qaamaadmaabaWaaubiaeqaoeqabaGaeyOeI0IaeqySde2aai WaaeaacqGHsislciGGSbGaai4BaiaacEgadaWadaqaaiaaigdacqGH sisldaWadaqaaiaaigdacqGHRaWkdaWcaaqaaiabeI7aXjaadIhaae aacqaH4oqCcqGHRaWkcaaIXaaaaaGaay5waiaaw2faamaavacabeqa beaacqGHsislcqaH4oqCcaWG4baabaGaamyzaaaaaiaawUfacaGLDb aaaiaawUhacaGL9baadaahaaqabeaacqaHYoGyaaaaneaacaWGLbaa aaGaay5waiaaw2faaaaaaaa@E03F@  (2.4.2)

Hence, the pdf of the minimum order statistic X (1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIfadaWgaa WcbaGaaiikaiaaigdacaGGPaaabeaaaaa@3908@  and maximum order statistic X (n) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIfadaWgaa WcbaGaaiikaiaad6gacaGGPaaabeaaaaa@3940@  of the WLnD are given by;

f 1:n (x)=n k=0 n1 (1) k ( n1 k ) [ αβ θ 2 ( 1+x ) e θx { log[ 1[ 1+ θx θ+1 ] e θx ] } β1 ( θ+1 )[ 1[ 1+ θx θ+1 ] e θx ] e α { log[ 1[ 1+ θx θ+1 ] e θx ] } β ]* [ e α { log[ 1[ 1+ θx θ+1 ] e θx ] } β ] k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaWaaubeaeqale aacaaIXaGaaiOoaiaad6gaaeqaneaacaWGMbaaaOGaaiikaiaadIha caGGPaGaeyypa0JaamOBamaaqahabaWaaubiaeqaleqabaGaam4Aaa qdbaGaaiikaiabgkHiTiaaigdacaGGPaaaaOWaaeWaaeaafaqabeGa baaabaGaamOBaiabgkHiTiaaigdaaeaacaWGRbaaaaGaayjkaiaawM caaaWcbaGaam4Aaiabg2da9iaaicdaaeaacaWGUbGaeyOeI0IaaGym aaqdcqGHris5aOWaamWaaeaadaWcaaqaaiabeg7aHjabek7aIjabeI 7aXnaaCaaaleqabaGaaGOmaaaakmaabmaabaGaaGymaiabgUcaRiaa dIhaaiaawIcacaGLPaaadaqfGaqabSqabeaacqGHsislcqaH4oqCca WG4baaneaacaWGLbaaaOWaaiWaaeaacqGHsislciGGSbGaai4Baiaa cEgadaWadaqaaiaaigdacqGHsisldaWadaqaaiaaigdacqGHRaWkda WcaaqaaiabeI7aXjaadIhaaeaacqaH4oqCcqGHRaWkcaaIXaaaaaGa ay5waiaaw2faamaavacabeWcbeqaaiabgkHiTiabeI7aXjaadIhaa0 qaaiaadwgaaaaakiaawUfacaGLDbaaaiaawUhacaGL9baadaahaaWc beqaaiabek7aIjabgkHiTiaaigdaaaaakeaadaqadaqaaiabeI7aXj abgUcaRiaaigdaaiaawIcacaGLPaaadaWadaqaaiaaigdacqGHsisl daWadaqaaiaaigdacqGHRaWkdaWcaaqaaiabeI7aXjaadIhaaeaacq aH4oqCcqGHRaWkcaaIXaaaaaGaay5waiaaw2faamaavacabeWcbeqa aiabgkHiTiabeI7aXjaadIhaa0qaaiaadwgaaaaakiaawUfacaGLDb aadaqfGaqabSqabeaacqaHXoqydaGadaqaaiabgkHiTiGacYgacaGG VbGaai4zamaadmaabaGaaGymaiabgkHiTmaadmaabaGaaGymaiabgU caRmaalaaabaGaeqiUdeNaamiEaaqaaiabeI7aXjabgUcaRiaaigda aaaacaGLBbGaayzxaaWaaubiaeqameqabaGaeyOeI0IaeqiUdeNaam iEaaGdbaGaamyzaaaaaSGaay5waiaaw2faaaGaay5Eaiaaw2haamaa CaaameqabaGaeqOSdigaaaqdbaGaamyzaaaaaaaakiaawUfacaGLDb aacaGGQaWaaubiaeqaleqabaGaam4AaaqdbaWaamWaaeaadaqfGaqa b4qabeaacqGHsislcqaHXoqydaGadaqaaiabgkHiTiGacYgacaGGVb Gaai4zamaadmaabaGaaGymaiabgkHiTmaadmaabaGaaGymaiabgUca RmaalaaabaGaeqiUdeNaamiEaaqaaiabeI7aXjabgUcaRiaaigdaaa aacaGLBbGaayzxaaWaaubiaeqabeqaaiabgkHiTiabeI7aXjaadIha aeaacaWGLbaaaaGaay5waiaaw2faaaGaay5Eaiaaw2haamaaCaaabe qaaiabek7aIbaaa0qaaiaadwgaaaaacaGLBbGaayzxaaaaaaaa@D249@  (2.4.3)

And

f n:n (x)=n[ αβ θ 2 ( 1+x ) e θx { log[ 1[ 1+ θx θ+1 ] e θx ] } β1 ( θ+1 )[ 1[ 1+ θx θ+1 ] e θx ] e α { log[ 1[ 1+ θx θ+1 ] e θx ] } β ] [ e α { log[ 1[ 1+ θx θ+1 ] e θx ] } β ] n1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaWaaubeaeqale aacaWGUbGaaiOoaiaad6gaaeqaneaacaWGMbaaaOGaaiikaiaadIha caGGPaGaeyypa0JaamOBamaadmaabaWaaSaaaeaacqaHXoqycqaHYo GycqaH4oqCdaahaaWcbeqaaiaaikdaaaGcdaqadaqaaiaaigdacqGH RaWkcaWG4baacaGLOaGaayzkaaWaaubiaeqaleqabaGaeyOeI0Iaeq iUdeNaamiEaaqdbaGaamyzaaaakmaacmaabaGaeyOeI0IaciiBaiaa c+gacaGGNbWaamWaaeaacaaIXaGaeyOeI0YaamWaaeaacaaIXaGaey 4kaSYaaSaaaeaacqaH4oqCcaWG4baabaGaeqiUdeNaey4kaSIaaGym aaaaaiaawUfacaGLDbaadaqfGaqabSqabeaacqGHsislcqaH4oqCca WG4baaneaacaWGLbaaaaGccaGLBbGaayzxaaaacaGL7bGaayzFaaWa aWbaaSqabeaacqaHYoGycqGHsislcaaIXaaaaaGcbaWaaeWaaeaacq aH4oqCcqGHRaWkcaaIXaaacaGLOaGaayzkaaWaamWaaeaacaaIXaGa eyOeI0YaamWaaeaacaaIXaGaey4kaSYaaSaaaeaacqaH4oqCcaWG4b aabaGaeqiUdeNaey4kaSIaaGymaaaaaiaawUfacaGLDbaadaqfGaqa bSqabeaacqGHsislcqaH4oqCcaWG4baaneaacaWGLbaaaaGccaGLBb GaayzxaaWaaubiaeqaleqabaGaeqySde2aaiWaaeaacqGHsislciGG SbGaai4BaiaacEgadaWadaqaaiaaigdacqGHsisldaWadaqaaiaaig dacqGHRaWkdaWcaaqaaiabeI7aXjaadIhaaeaacqaH4oqCcqGHRaWk caaIXaaaaaGaay5waiaaw2faamaavacabeadbeqaaiabgkHiTiabeI 7aXjaadIhaa4qaaiaadwgaaaaaliaawUfacaGLDbaaaiaawUhacaGL 9baadaahaaadbeqaaiabek7aIbaaa0qaaiaadwgaaaaaaaGccaGLBb GaayzxaaWaaubiaeqaleqabaGaamOBaiabgkHiTiaaigdaa0qaamaa dmaabaWaaubiaeqaoeqabaGaeyOeI0IaeqySde2aaiWaaeaacqGHsi slciGGSbGaai4BaiaacEgadaWadaqaaiaaigdacqGHsisldaWadaqa aiaaigdacqGHRaWkdaWcaaqaaiabeI7aXjaadIhaaeaacqaH4oqCcq GHRaWkcaaIXaaaaaGaay5waiaaw2faamaavacabeqabeaacqGHsisl cqaH4oqCcaWG4baabaGaamyzaaaaaiaawUfacaGLDbaaaiaawUhaca GL9baadaahaaqabeaacqaHYoGyaaaaneaacaWGLbaaaaGaay5waiaa w2faaaaaaaa@C275@  (2.4.4)

,p>respectively.

Parameter estimation via maximum likelihood

Let X 1 , X 2 ,...., X n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaBa aaleaacaaIXaaabeaakiaacYcacaWGybWaaSbaaSqaaiaaikdaaeqa aOGaaiilaiaac6cacaGGUaGaaiOlaiaac6cacaGGSaGaamiwamaaBa aaleaacaWGUbaabeaaaaa@41E8@ > be a sample of size ‘n’ independently and identically distributed random variables from the WLnD with unknown parameters α, β and Ө defined previously. The pdf of the WLnD is given from (2.1.3) as

f( x )= αβ θ 2 ( 1+x ) e θx { log[ 1[ 1+ θx θ+1 ] e θx ] } β1 e α { log[ 1[ 1+ θx θ+1 ] e θx ] } β ( θ+1 )[ 1[ 1+ θx θ+1 ] e θx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm aabaGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqySdeMa eqOSdiMaeqiUde3aaWbaaSqabeaacaaIYaaaaOWaaeWaaeaacaaIXa Gaey4kaSIaamiEaaGaayjkaiaawMcaamaavacabeWcbeqaaiabgkHi TiabeI7aXjaadIhaa0qaaiaadwgaaaGcdaGadaqaaiabgkHiTiGacY gacaGGVbGaai4zamaadmaabaGaaGymaiabgkHiTmaadmaabaGaaGym aiabgUcaRmaalaaabaGaeqiUdeNaamiEaaqaaiabeI7aXjabgUcaRi aaigdaaaaacaGLBbGaayzxaaWaaubiaeqaleqabaGaeyOeI0IaeqiU deNaamiEaaqdbaGaamyzaaaaaOGaay5waiaaw2faaaGaay5Eaiaaw2 haamaaCaaaleqabaGaeqOSdiMaeyOeI0IaaGymaaaakmaavacabeWc beqaaiabgkHiTiabeg7aHnaacmaabaGaeyOeI0IaciiBaiaac+gaca GGNbWaamWaaeaacaaIXaGaeyOeI0YaamWaaeaacaaIXaGaey4kaSYa aSaaaeaacqaH4oqCcaWG4baabaGaeqiUdeNaey4kaSIaaGymaaaaai aawUfacaGLDbaadaqfGaqabWqabeaacqGHsislcqaH4oqCcaWG4baa oeaacaWGLbaaaaWccaGLBbGaayzxaaaacaGL7bGaayzFaaWaaWbaaW qabeaacqaHYoGyaaaaneaacaWGLbaaaaGcbaWaaeWaaeaacqaH4oqC cqGHRaWkcaaIXaaacaGLOaGaayzkaaWaamWaaeaacaaIXaGaeyOeI0 YaamWaaeaacaaIXaGaey4kaSYaaSaaaeaacqaH4oqCcaWG4baabaGa eqiUdeNaey4kaSIaaGymaaaaaiaawUfacaGLDbaadaqfGaqabSqabe aacqGHsislcqaH4oqCcaWG4baaneaacaWGLbaaaaGccaGLBbGaayzx aaaaaaaa@9B4F@

The likelihood function is given by;

L( X _ |α,β,θ )= ( α β α θ 2 ) n i=1 n { ( 1+ x i ) e θ x i } i=1 n { log[ 1[ 1+ θ x i θ+1 ] e θ x i ] } β1 ( θ+1 ) n i=1 n { 1[ 1+ θ x i θ+1 ] e θ x i } e α i=1 n { log[ 1[ 1+ θ x i θ+1 ] e θ x i ] } β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamitamaabm aabaGabmiwayaaDaGaaiiFaiabeg7aHjaacYcacqaHYoGycaGGSaGa eqiUdehacaGLOaGaayzkaaGaeyypa0ZaaSaaaeaadaqfGaqabSqabe aacaWGUbaaneaadaqadaqaaiabeg7aHjabek7aInaaCaaaoeqabaGa eqySdegaa0GaeqiUde3aaWbaa4qabeaacaaIYaaaaaqdcaGLOaGaay zkaaaaaOWaaebCaeaadaGadaqaamaabmaabaGaaGymaiabgUcaRiaa dIhadaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaadaqfGaqabS qabeaacqGHsislcqaH4oqCcaWG4bWaaSbaaWqaaiaadMgaaeqaaaqd baGaamyzaaaaaOGaay5Eaiaaw2haaaWcbaGaamyAaiabg2da9iaaig daaeaacaWGUbaaniabg+GivdGcdaqeWbqaamaacmaabaGaeyOeI0Ia ciiBaiaac+gacaGGNbWaamWaaeaacaaIXaGaeyOeI0YaamWaaeaaca aIXaGaey4kaSYaaSaaaeaacqaH4oqCcaWG4bWaaSbaaSqaaiaadMga aeqaaaGcbaGaeqiUdeNaey4kaSIaaGymaaaaaiaawUfacaGLDbaada qfGaqabSqabeaacqGHsislcqaH4oqCcaWG4bWaaSbaaWqaaiaadMga aeqaaaqdbaGaamyzaaaaaOGaay5waiaaw2faaaGaay5Eaiaaw2haam aaCaaaleqabaGaeqOSdiMaeyOeI0IaaGymaaaaaeaacaWGPbGaeyyp a0JaaGymaaqaaiaad6gaa0Gaey4dIunaaOqaamaabmaabaGaeqiUde Naey4kaSIaaGymaaGaayjkaiaawMcaamaaCaaaleqabaGaamOBaaaa kmaarahabaWaaiWaaeaacaaIXaGaeyOeI0YaamWaaeaacaaIXaGaey 4kaSYaaSaaaeaacqaH4oqCcaWG4bWaaSbaaSqaaiaadMgaaeqaaaGc baGaeqiUdeNaey4kaSIaaGymaaaaaiaawUfacaGLDbaadaqfGaqabS qabeaacqGHsislcqaH4oqCcaWG4bWaaSbaaWqaaiaadMgaaeqaaaqd baGaamyzaaaaaOGaay5Eaiaaw2haamaavacabeWcbeqaaiabeg7aHn aaqahabaWaaiWaaeaacqGHsislciGGSbGaai4BaiaacEgadaWadaqa aiaaigdacqGHsisldaWadaqaaiaaigdacqGHRaWkdaWcaaqaaiabeI 7aXjaadIhadaWgaaadbaGaamyAaaqabaaaleaacqaH4oqCcqGHRaWk caaIXaaaaaGaay5waiaaw2faamaavacabeadbeqaaiabgkHiTiabeI 7aXjaadIhadaWgaaqaaiaadMgaaeqaaaGdbaGaamyzaaaaaSGaay5w aiaaw2faaaGaay5Eaiaaw2haamaaCaaameqabaGaeqOSdigaaaqaai aadMgacqGH9aqpcaaIXaaabaGaamOBaaGdcqGHris5aaqdbaGaamyz aaaaaSqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHpis1aa aaaaa@CA34@  (2.5.1)

Let the log-likelihood function, l=logL( X _ |α,β,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamiBaiabg2 da9iGacYgacaGGVbGaai4zaiaadYeadaqadaqaaiqadIfagaqhaiaa cYhacqaHXoqycaGGSaGaeqOSdiMaaiilaiabeI7aXbGaayjkaiaawM caaaaa@46EF@ , therefore

l=nlogα+nlogβ+2nlogθnlog( θ+1 )+ i=1 n log( 1+ x i ) θ i=1 n x i +( β1 ) i=1 n log{ log[ 1[ 1+ θ x i θ+1 ] e θ x i ] } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaamiBaiabg2 da9iaad6gaciGGSbGaai4BaiaacEgacqaHXoqycqGHRaWkcaWGUbGa ciiBaiaac+gacaGGNbGaeqOSdiMaey4kaSIaaGOmaiaad6gaciGGSb Gaai4BaiaacEgacqaH4oqCcqGHsislcaWGUbGaciiBaiaac+gacaGG NbWaaeWaaeaacqaH4oqCcqGHRaWkcaaIXaaacaGLOaGaayzkaaGaey 4kaSYaaabCaeaaciGGSbGaai4BaiaacEgadaqadaqaaiaaigdacqGH RaWkcaWG4bWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaaale aacaWGPbGaeyypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoakiabgkHi TiabeI7aXnaaqahabaGaamiEamaaBaaaleaacaWGPbaabeaaaeaaca WGPbGaeyypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoakiabgUcaRmaa bmaabaGaeqOSdiMaeyOeI0IaaGymaaGaayjkaiaawMcaamaaqahaba GaciiBaiaac+gacaGGNbWaaiWaaeaacqGHsislciGGSbGaai4Baiaa cEgadaWadaqaaiaaigdacqGHsisldaWadaqaaiaaigdacqGHRaWkda WcaaqaaiabeI7aXjaadIhadaWgaaWcbaGaamyAaaqabaaakeaacqaH 4oqCcqGHRaWkcaaIXaaaaaGaay5waiaaw2faamaavacabeWcbeqaai abgkHiTiabeI7aXjaadIhadaWgaaadbaGaamyAaaqabaaaneaacaWG LbaaaaGccaGLBbGaayzxaaaacaGL7bGaayzFaaaaleaacaWGPbGaey ypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoaaaa@97FB@

i=1 n log( [ 1[ 1+ θ x i θ+1 ] e θ x i ] ) α i=1 n { log[ 1[ 1+ θ x i θ+1 ] e θ x i ] } β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Yaaa bCaeaaciGGSbGaai4BaiaacEgadaqadaqaamaadmaabaGaaGymaiab gkHiTmaadmaabaGaaGymaiabgUcaRmaalaaabaGaeqiUdeNaamiEam aaBaaaleaacaWGPbaabeaaaOqaaiabeI7aXjabgUcaRiaaigdaaaaa caGLBbGaayzxaaWaaubiaeqaleqabaGaeyOeI0IaeqiUdeNaamiEam aaBaaameaacaWGPbaabeaaa0qaaiaadwgaaaaakiaawUfacaGLDbaa aiaawIcacaGLPaaaaSqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaa qdcqGHris5aOGaeyOeI0IaeqySde2aaabCaeaadaGadaqaaiabgkHi TiGacYgacaGGVbGaai4zamaadmaabaGaaGymaiabgkHiTmaadmaaba GaaGymaiabgUcaRmaalaaabaGaeqiUdeNaamiEamaaBaaaleaacaWG PbaabeaaaOqaaiabeI7aXjabgUcaRiaaigdaaaaacaGLBbGaayzxaa WaaubiaeqaleqabaGaeyOeI0IaeqiUdeNaamiEamaaBaaameaacaWG Pbaabeaaa0qaaiaadwgaaaaakiaawUfacaGLDbaaaiaawUhacaGL9b aadaahaaWcbeqaaiabek7aIbaaaeaacaWGPbGaeyypa0JaaGymaaqa aiaad6gaa0GaeyyeIuoaaaa@7B9C@  (2.5.2)

Differentiating l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqadeaabaGaciaacaqabeaadaqaaqaaaOqaaGqadabaaaaaaa aapeGaa8hBaaaa@3945@  partially with respect to α, β and Ө respectively gives;

l α = n α i=1 n { log[ 1[ 1+ θ x i θ+1 ] e θ x i ] } β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHciITcaWGSbaabaGaeyOaIyRaeqySdegaaiabg2da9maalaaabaGa amOBaaqaaiabeg7aHbaacqGHsisldaaeWbqaamaacmaabaGaeyOeI0 IaciiBaiaac+gacaGGNbWaamWaaeaacaaIXaGaeyOeI0YaamWaaeaa caaIXaGaey4kaSYaaSaaaeaacqaH4oqCcaWG4bWaaSbaaSqaaiaadM gaaeqaaaGcbaGaeqiUdeNaey4kaSIaaGymaaaaaiaawUfacaGLDbaa daqfGaqabSqabeaacqGHsislcqaH4oqCcaWG4bWaaSbaaWqaaiaadM gaaeqaaaqdbaGaamyzaaaaaOGaay5waiaaw2faaaGaay5Eaiaaw2ha amaaCaaaleqabaGaeqOSdigaaaqaaiaadMgacqGH9aqpcaaIXaaaba GaamOBaaqdcqGHris5aaaa@6375@  (2.5.3)

l β = n β + i=1 n log{ log[ 1[ 1+ θ x i θ+1 ] e θ x i ] } α i=1 n { log[ 1[ 1+ θ x i θ+1 ] e θ x i ] } β { log[ 1[ 1+ θ x i θ+1 ] e θ x i ] } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHciITcaWGSbaabaGaeyOaIyRaeqOSdigaaiabg2da9maalaaabaGa amOBaaqaaiabek7aIbaacqGHRaWkdaaeWbqaaiGacYgacaGGVbGaai 4zamaacmaabaGaeyOeI0IaciiBaiaac+gacaGGNbWaamWaaeaacaaI XaGaeyOeI0YaamWaaeaacaaIXaGaey4kaSYaaSaaaeaacqaH4oqCca WG4bWaaSbaaSqaaiaadMgaaeqaaaGcbaGaeqiUdeNaey4kaSIaaGym aaaaaiaawUfacaGLDbaadaqfGaqabSqabeaacqGHsislcqaH4oqCca WG4bWaaSbaaWqaaiaadMgaaeqaaaqdbaGaamyzaaaaaOGaay5waiaa w2faaaGaay5Eaiaaw2haaaWcbaGaamyAaiabg2da9iaaigdaaeaaca WGUbaaniabggHiLdGccqGHsislcqaHXoqydaaeWbqaamaacmaabaGa eyOeI0IaciiBaiaac+gacaGGNbWaamWaaeaacaaIXaGaeyOeI0Yaam WaaeaacaaIXaGaey4kaSYaaSaaaeaacqaH4oqCcaWG4bWaaSbaaSqa aiaadMgaaeqaaaGcbaGaeqiUdeNaey4kaSIaaGymaaaaaiaawUfaca GLDbaadaqfGaqabSqabeaacqGHsislcqaH4oqCcaWG4bWaaSbaaWqa aiaadMgaaeqaaaqdbaGaamyzaaaaaOGaay5waiaaw2faaaGaay5Eai aaw2haamaaCaaaleqabaGaeqOSdigaaOWaaiWaaeaacqGHsislciGG SbGaai4BaiaacEgadaWadaqaaiaaigdacqGHsisldaWadaqaaiaaig dacqGHRaWkdaWcaaqaaiabeI7aXjaadIhadaWgaaWcbaGaamyAaaqa baaakeaacqaH4oqCcqGHRaWkcaaIXaaaaaGaay5waiaaw2faamaava cabeWcbeqaaiabgkHiTiabeI7aXjaadIhadaWgaaadbaGaamyAaaqa baaaneaacaWGLbaaaaGccaGLBbGaayzxaaaacaGL7bGaayzFaaaale aacaWGPbGaeyypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoaaaa@A379@  (2.5.4)

l θ = 2n θ n θ+1 i=1 n x i +( β1 ) i=1 n { x i e θ x i [ 1 ( θ+1 ) 2 θ x i +θ+1 θ+1 ] { log[ 1[ 1+ θ x i θ+1 ] e θ x i ] }{ [ 1[ 1+ θ x i θ+1 ] e θ x i ] } } i=1 n { x i e θ x i [ 1 ( θ+1 ) 2 θ x i +θ+1 θ+1 ] ( [ 1[ 1+ θ x i θ+1 ] e θ x i ] ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHciITcaWGSbaabaGaeyOaIyRaeqiUdehaaiabg2da9maalaaabaGa aGOmaiaad6gaaeaacqaH4oqCaaGaeyOeI0YaaSaaaeaacaWGUbaaba GaeqiUdeNaey4kaSIaaGymaaaacqGHsisldaaeWbqaaiaadIhadaWg aaWcbaGaamyAaaqabaaabaGaamyAaiabg2da9iaaigdaaeaacaWGUb aaniabggHiLdGccqGHRaWkdaqadaqaaiabek7aIjabgkHiTiaaigda aiaawIcacaGLPaaadaaeWbqaamaacmaabaWaaSaaaeaacaWG4bWaaS baaSqaaiaadMgaaeqaaOWaaubiaeqaleqabaGaeyOeI0IaeqiUdeNa amiEamaaBaaameaacaWGPbaabeaaa0qaaiaadwgaaaGcdaWadaqaam aaleaaleaacaaIXaaabaWaaeWaaeaacqaH4oqCcqGHRaWkcaaIXaaa caGLOaGaayzkaaWaaWbaaWqabeaacaaIYaaaaaaakiabgkHiTmaale aaleaacqaH4oqCcaWG4bWaaSbaaWqaaiaadMgaaeqaaSGaey4kaSIa eqiUdeNaey4kaSIaaGymaaqaaiabeI7aXjabgUcaRiaaigdaaaaaki aawUfacaGLDbaaaeaadaGadaqaaiabgkHiTiGacYgacaGGVbGaai4z amaadmaabaGaaGymaiabgkHiTmaadmaabaGaaGymaiabgUcaRmaala aabaGaeqiUdeNaamiEamaaBaaaleaacaWGPbaabeaaaOqaaiabeI7a XjabgUcaRiaaigdaaaaacaGLBbGaayzxaaWaaubiaeqaleqabaGaey OeI0IaeqiUdeNaamiEamaaBaaameaacaWGPbaabeaaa0qaaiaadwga aaaakiaawUfacaGLDbaaaiaawUhacaGL9baadaGadaqaamaadmaaba GaaGymaiabgkHiTmaadmaabaGaaGymaiabgUcaRmaalaaabaGaeqiU deNaamiEamaaBaaaleaacaWGPbaabeaaaOqaaiabeI7aXjabgUcaRi aaigdaaaaacaGLBbGaayzxaaWaaubiaeqaleqabaGaeyOeI0IaeqiU deNaamiEamaaBaaameaacaWGPbaabeaaa0qaaiaadwgaaaaakiaawU facaGLDbaaaiaawUhacaGL9baaaaaacaGL7bGaayzFaaaaleaacaWG PbGaeyypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoakiabgkHiTmaaqa habaWaaiWaaeaadaWcaaqaaiaadIhadaWgaaWcbaGaamyAaaqabaGc daqfGaqabSqabeaacqGHsislcqaH4oqCcaWG4bWaaSbaaWqaaiaadM gaaeqaaaqdbaGaamyzaaaakmaadmaabaWaaSqaaSqaaiaaigdaaeaa daqadaqaaiabeI7aXjabgUcaRiaaigdaaiaawIcacaGLPaaadaahaa adbeqaaiaaikdaaaaaaOGaeyOeI0YaaSqaaSqaaiabeI7aXjaadIha daWgaaadbaGaamyAaaqabaWccqGHRaWkcqaH4oqCcqGHRaWkcaaIXa aabaGaeqiUdeNaey4kaSIaaGymaaaaaOGaay5waiaaw2faaaqaamaa bmaabaWaamWaaeaacaaIXaGaeyOeI0YaamWaaeaacaaIXaGaey4kaS YaaSaaaeaacqaH4oqCcaWG4bWaaSbaaSqaaiaadMgaaeqaaaGcbaGa eqiUdeNaey4kaSIaaGymaaaaaiaawUfacaGLDbaadaqfGaqabSqabe aacqGHsislcqaH4oqCcaWG4bWaaSbaaWqaaiaadMgaaeqaaaqdbaGa amyzaaaaaOGaay5waiaaw2faaaGaayjkaiaawMcaaaaaaiaawUhaca GL9baaaSqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHris5 aaaa@E7DA@

+αβ i=1 n { x i e θ x i [ 1 ( θ+1 ) 2 θ x i +θ+1 θ+1 ] { log[ 1[ 1+ θ x i θ+1 ] e θ x i ] } β1 [ 1[ 1+ θ x i θ+1 ] e θ x i ] } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaey4kaSIaeq ySdeMaeqOSdi2aaabCaeaadaGadaqaamaalaaabaGaamiEamaaBaaa leaacaWGPbaabeaakmaavacabeWcbeqaaiabgkHiTiabeI7aXjaadI hadaWgaaadbaGaamyAaaqabaaaneaacaWGLbaaaOWaamWaaeaadaWc baWcbaGaaGymaaqaamaabmaabaGaeqiUdeNaey4kaSIaaGymaaGaay jkaiaawMcaamaaCaaameqabaGaaGOmaaaaaaGccqGHsisldaWcbaWc baGaeqiUdeNaamiEamaaBaaameaacaWGPbaabeaaliabgUcaRiabeI 7aXjabgUcaRiaaigdaaeaacqaH4oqCcqGHRaWkcaaIXaaaaaGccaGL BbGaayzxaaWaaiWaaeaacqGHsislciGGSbGaai4BaiaacEgadaWada qaaiaaigdacqGHsisldaWadaqaaiaaigdacqGHRaWkdaWcaaqaaiab eI7aXjaadIhadaWgaaWcbaGaamyAaaqabaaakeaacqaH4oqCcqGHRa WkcaaIXaaaaaGaay5waiaaw2faamaavacabeWcbeqaaiabgkHiTiab eI7aXjaadIhadaWgaaadbaGaamyAaaqabaaaneaacaWGLbaaaaGcca GLBbGaayzxaaaacaGL7bGaayzFaaWaaWbaaSqabeaacqaHYoGycqGH sislcaaIXaaaaaGcbaWaamWaaeaacaaIXaGaeyOeI0YaamWaaeaaca aIXaGaey4kaSYaaSaaaeaacqaH4oqCcaWG4bWaaSbaaSqaaiaadMga aeqaaaGcbaGaeqiUdeNaey4kaSIaaGymaaaaaiaawUfacaGLDbaada qfGaqabSqabeaacqGHsislcqaH4oqCcaWG4bWaaSbaaWqaaiaadMga aeqaaaqdbaGaamyzaaaaaOGaay5waiaaw2faaaaaaiaawUhacaGL9b aaaSqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHris5aaaa @932F@  (2.5.5)

Equating equations (2.5.3), (2.5.4) and (2.5.5) to zero and solving for the solution of the non-linear system of equations will give us the maximum likelihood estimates of parameters α,β&θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbmqaaeaaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaai ilaiabek7aIjaacAcacqaH4oqCaaa@3DC7@  respectively. However, the above equations cannot be solved manually due to their complexity unless numerically with the help of statistically inclined computer programs like Python, R, SAS, etc., when data sets are available.

Results and discussions

This section presents four datasets, their descriptive statistics, graphics and applications to some selected extensions of the Lindley distribution including the classical Lindley distribution. We have compared the performance of the Weibull-Lindley distribution (WLlD) to some families of Lindley distribution such as Lomax-Lindley distribution (LLlD), Two-parameter Lindley distribution (TPLlD), Transmuted Lindley distribution (TLlD) and the Lindley distribution (LlD).

 Data sets and their nature

In this section, four different datasets and their summary are presented for fitting the above listed distributions. The available data sets I, II, III, and IV and their respective summary statistics are provided in Table 1–5 respectively as follows;

Parameters

n

Minimum

Median

Mean

Maximum

Variance

Skewness

Kurtosis

Values

20

1.1

1.475

1.7

2.05

1.9

4.1

0.4958

1.8625

7.1854

Table 1 Summary statistics for dataset I

Parameters

n

Minimum

Median

Mean

Maximum

Variance

Skewness

Kurtosis

Values

20

40

86.75

119

140.8

113.45

165

1280.892

-0.3552

-0.89

Table 2 Summary statistics for dataset II

Parameters

n

Minimum

Median

Mean

Maximum

Variance

Skewness

Kurtosis

Data set I

63

0.55

1.375

1.59

1.685

1.507

2.24

0.105

-0.8786

3.9238

Table 3 Summary Statistics for data set III

Parameters

n

Minimum

Median

Mean

Maximum

Variance

Skewness

Kurtosis

Values

59

4.1

8.45

10.6

16.85

13.49

39.2

64.8266

1.6083

2.256

Table 4 Descriptive statistics for dataset IV

Parameter estimates

ƖƖ=(log-likelihood value)

AIC

A*

W*

K-S

P-Value

Ranks

(K-S)

0.5842

16.0004

38.0009

0.2483

0.0428

0.167

0.6324

1

1.2337

3.8112

1.1589

24.9726

53.9452

0.6295

0.1063

0.3113

0.0414

2

-0.9888

0.8926

27.2805

58.561

0.6401

0.108

0.2885

0.0716

3

9.7008

0.8162

30.2496

62.4991

0.6758

0.1141

0.3911

0.0044

4

0.361

29.8421

65.6841

0.6552

0.1107

0.416

0.002

5

9.6021

2.4483

Table 5 The statistics ll, AIC, A*, W* and K-S for the fitted models to the first dataset

Dataset I: This dataset represents the lifetime’s data relating to relief times (in minutes) of 20 patients receiving an analgesic and reported by Gross et al.,34 and has been used by Shanker et al.,35 Table 1.

Dataset II: This data represent the survival times in weeks for male rats Lawless et al.,36 (Table 2).

Data set III: This data set represents the strength of 1.5cm glass fibers initially collected by members of staff at the UK national laboratory. It has been used by Bourguignon et al.,18 Afify et al.,37  Barreto Souza et al.,38 Oguntunde et al.,39 Ieren  et al.,40 as well as Smith et al.,41 Its summary is given as follows: (Table 3).

Dataset IV: This dataset represents 59 observations of the monthly actual taxes revenue in Egypt (in 1,000 million Egyptian pounds) between January 2006 and November 2010. The data has been previously used by Owoloko et al.42 The descriptive statistics for this data are as follows:

We also provide some histograms and densities for the three data sets as shown in Figure 5–8 below respectively (Table 4).

From the summary statistics of the four data sets, we found that data sets I and IV are positively skewed, while II is a bit negatively skewed or approximately normal and III is negatively skewed. Also, data sets I, III and IV have higher kurtosis while II have low level or degree of peakness.

Figure 5 Histogram and density plot for the Relief times of 20 patients (Data set I).

Figure 6 Histogram and density plot for the survival times in weeks for male rats (Data set II).

Figure 7 A histogram and density plot for the strength of 1.5cm glass fibres (Data set III).

Figure 8 A histogram and density plot for the monthly actual taxes revenue in Egypt (Data set IV).

Analysis of data

These four different datasets presented above were used to fit all the above listed Lindley distributions by applying the formulas of the test statistics in section 2 in order to get the best fitted model and the results are presented as follow in the four tables for each dataset below: (Table 5).

From Table 5, the values of the parameter MLEs and the corresponding values of ƖƖ, AIC, A*, W* and K-S for each model show that the Weibull-Lindley distribution (WLlD) has better performance compared to the other four models namely: Lomax-Lindley distribution (LLlD), Two-parameter Lindley distribution (TPLlD), Transmuted Lindley distribution (TLlD) and the Lindley distribution (LlD) and hence becomes the best fitted distribution based the data set I (Table 6).

Parameter estimates

ƖƖ=(log-likelihood value)

AIC

A*

W*

K-S

P-Value

Ranks

(K-S)

0.0278

106.6467

219.2935

0.4311

0.062

0.3225

0.0313

1

1.6862

1.0152

0.0198

112.8891

231.7781

0.5199

0.076

0.3772

0.0068

2

1.9445

2.8037

0.0371

128.3464

260.6929

0.4103

0.0586

0.6941

8.57E-09

3

0.3054

1.9987

4442.078

8888.156

NaN

NaN

1

<2.2e-16

5

0.6326

2.2161

4926.226

9854.452

NaN

NaN

1

<2.2e-16

4

Table 6 The statistics ll, AIC, A*, W* and K-S for the fitted models to the second dataset

Again the results in Table 6 above shows that the Weibull-Lindley distribution (WLlD) fits the second dataset better than the other four models (LLlD, TPLlD, TLlD and LlD) because the values of the statistics; AIC, A*, W* and K-S are smaller for the WLlD than the other models and therefore it is considered as the best fitted distribution based the data set II (Table 7).

Distributions

Parameter estimates

ƖƖ=(log-likelihood value)

AIC

A*

W*

K-S

P-Value

Ranks

(K-S)

1.4523

34.1708

74.3416

4.2254

0.7768

0.224

0.0033

1

7.6251

1.7248

0.361

81.4714

168.9428

3.074

0.5619

0.3213

3.60E-06

2

9.6021

2.4483

1.391

63.8482

131.6963

3.1901

0.5833

0.3413

6.70E-07

3

-0.9937

1.2155

71.0355

146.071

3.1414

0.5744

0.3427

5.90E-07

4

9.2573

0.9957

82.5853

167.1706

3.0788

0.563

0.3885

8.20E-09

5

Table 7 The statistics ll, AIC, A*, W* and K-S for the fitted models to the third dataset

The results from Table 7 also agrees with the previous results that the WLlD is more flexible compared to the three other models this also agrees with the fact that generalizing any continuous distribution provides a compound distribution with at least better fit than the classical distribution (i.e Lindley) irrespective of the nature of the data used provide it is asymmetry Table 8.

Distributions

Parameter estimates

ƖƖ=(log-likelihood value)

AIC

A*

W*

K-S

P-Value

Ranks

(K-S)

0.3767

199.8163

405.6327

0.7927

0.1329

0.1597

0.0986

1

8.5414

0.8256

0.0809

201.0534

408.1067

1.127

0.1849

0.1558

0.1139

2

6.6768

3.1649

0.1429

199.6626

403.3251

1.4251

0.229

0.1676

0.0728

5

-0.4154

0.1618

199.324

402.648

1.2489

0.2007

0.2084

0.0119

3

4.938

0.1361

200.6599

403.3198

1.2999

0.2087

0.1844

0.0361

4

Table 8 The statistics ll, AIC, A*, W* and K-S for the fitted models to the fourth dataset

Lastly, our results in Table 8 provides the same results as obtained in the above previous tables with the Weibull-Lindley distribution performing better than the other three distributions considered in this study.

The following figures displayed the histogram and estimated densities of the fitted models for the four real life data sets used in this study.

From the estimated density plots in Figures 9 we can observe that though there is no big difference between the performance of the other four models, it is very clear that the performance of the Weibull-Lindley distribution (WLlD) remains the best and consistent irrespective of the nature the datasets as compared to the Lomax-Lindley distribution (LLlD), Two-parameter Lindley distribution (TPLlD), Transmuted Lindley distribution (TLlD) and the Lindley distribution (LlD).

Figure 9 Histogram and estimated densities of the Weibull-Lindley distribution (WLlD), Lomax-Lindley distribution (LLlD), Two-parameter Lindley distribution (TPLlD), Transmuted Lindley distribution (TLlD) and the Lindley distribution (LlD) for the four real life datasets (dataset I, II, III and IV).

Furthermore, the performance of the Weibull-Lindley could be attributed to the fact that the Weibull-Lindley distribution is heavy-tailed and highly skewed to the right with excellent flexibility which allows it to take various shapes depending on the parameter values and it also exhibit some degree of kurtosis all of which are features of the four datasets used in this research, hence, the Weibull-Lindley distribution will be more appropriate for lifetime datasets which are positively skewed with a higher degree of peakness as well as those that are approximately normal with observations above zero.

Hence, having demonstrated earlier in Tables 5–8, we have a similar conclusion based on figure 3-5 that the Weibull-Lindley distribution has a better fit for the four data sets considered in this study.

Summary conclusion

This article proposed a new distribution called Weibull-Lindley distribution. The Mathematical and Statistical properties of the new distribution have been derived and studied extensively. Some of its properties with graphical analysis and discussion on their usefulness and applications were also considered. The model parameters were estimated using maximum likelihood method and we have a conclusion based on our applications of the model to four real life datasets that the new distribution (WLnD) has a better fit compared to the other four already existing models considered in this study.

Acknowledgements

None.

Conflict of interest

Author declares that there is no conflicts of interest.

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