# A New Scale-Invariant Lindley Extension Distribution and Its Applications

Authors: Mohamed Kayid; Rayof Alskhabrah; Arwa M. Alshangiti
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

A new scale-invariant extension of the Lindley distribution and its power generalization has been introduced. The moments and the moment-generating functions of the proposed models have closed forms. The failure rate, the mean residual life, and theα-quantile residual life functions have been explored. The failure rate function of these models accommodates increasing, bathtub-shaped, and increasing then bathtub-shaped forms. The parameters of the models have been estimated by the maximum likelihood method for the complete and right-censored data. In a simulation study, the efficiency and consistency of the maximum likelihood estimator have been investigated. Then, the proposed models were fitted to four data sets to show their flexibility and applicability.


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## Body

## 1. Introduction

The Lindley distribution introduced by Lindley [1] has been attracted much interest in recent years. The probability density function (PDF) and the cumulative distribution function (CDF) of the Lindley are, respectively, as follows:(1)fx=θ21+θ1+xe−θx,θ>0,x≥0,Fx=1−1+θ+θx1+θe−θx,θ>0,x≥0.Clearly the Lindley distribution is a mixture of two gamma distributionsG1,θ, gamma with shape parameter 1 and scale θ and G2,θ with weights θ/1+θ and 1/1+θ, respectively.Ghitany et al. [2] explored some properties of the Lindley model. Many authors introduced generalizations of this model or applied it in their models. Among them, Shanker and Mishra [3], Shanker and Ghebretsadik [4], Merovci and Sharma [5], Zakerzade and Dolati [6], Ibrahim et al. [7], and Shanker et al. [8] introduced generalizations of Lindley with some more parameters. Moreover, Sankaran [9], Zamani and Ismail [10], Ghitany et al. [11], Al-Mutairi et al. [12], Al-Babtain et al. [13], Ghitany et al. [14], Al-Mutairi et al. [15], Abouammoh et al. [16], Ibrahim et al. [17], Marthin and Rao [18], Al-Babtain et al. [19], Joshi and Kumar [20], Afify et al. [21], Chesneau et al. [22], Algarni [23], and many others used the Lindley distribution in their research or extended it. These generalized models are used in the reliability theory and are useful for describing data from different sources, e.g., lifetime of an object, strength of a material, and so on.One distribution familyfx is said to be scale invariant if fx times the transformation Jacobian remains unchanged after transforming x to αx (scale transform). So, changing the scale or units of x leaves the fit invariant. Shanker and Mishra [3] proposed one scale-invariant quasi Lindley distribution. Unfortunately, the Lindley model and some of its generalizations do not satisfy this property. The aim of this paper is to introduce one scale invariant extension of the Lindley distribution.In Sections2 and 3, the new distributions are defined, and some of their properties have been studied. In Section 4, the estimation of the parameters has been discussed. Section 5 is devoted to investigation of the maximum likelihood estimator (MLE) of the parameters and its behaviour by simulation. In Section 6, four data sets have been analyzed by the proposed distributions to show their applicability.


## 2. New Lindley Extension

We introduce one extended Lindley (EL) distribution with parametersk>0, m>0, k≠m, δ>0, and θ>0, namely, ELk,m;δ,θ, with the PDF as follows:(2)fx=θΓm+δδθxk−1Γk+θxm−1e−θx,x≥0.It is a mixture of two gamma distributionsGm,θ and Gk,θ with weights Γm/Γm+δ and δ/Γm+δ, respectively. In applications, when we want to fit this model to one data set, we may consider k and m as integers (discrete), e.g., k,m=1,2,…. So, we can consider a suitable sequence of pairs k,m and search for a good model by estimating δ,θ. In this way, we avoid headaches of optimizing four dimensional functions.Form=2, k=1, and δ=θ, it is the Lindley model. Figure 1 shows the PDF for some various parameters and indicates that the PDF accommodates decreasing, unimodal, and biomodal forms.Figure 1
The PDF ofELk,m;δ,θ for some values of parameters.
(a)(b)It is straightforward to show that the CDF ofELk,m;δ,θ is(3)Fx=1−1Γm+δδΓkΓθx,k+Γθx,m,x≥0,in which Γx,α=∫x∞tα−1e−tdt is the upper incomplete gamma function. Let X∼ELk,m;δ,θ, then the rth moment is(4)EXr=1θr1Γm+δδΓk+rΓk+Γm+r.It is clear that whenk>1 and m>1, EXr>1/θr. The moment-generating function can also be simplified to(5)EetX=1Γm+δθθ−tmΓm+δθθ−tk.


### 2.1. Dynamic Measures

One of the most elementary measures in reliability theory and survival analysis is the reliability function,Rx=1−Fx. Moreover, dynamic measures like the failure rate function, the mean residual life (MRL) function, and the α-quantile residual life (α-QRL) function play key role in investigation the behaviour of lifetime. Let X∼ELk,m;δ,θ, then the failure rate function is(6)hx=fxRx=Γkθxm−1+δθxk−1ΓkΓθx,m+δΓθx,kθe−θx.Applying Glaser’s technique, this failure rate function may exhibit increasing, bathtub-shaped, or increasing then bathtub-shaped forms (see Glaser [24] or Theorem 2.8 in the study by Lai and Xie [25]). Figure 2 draws the failure rate function for some values of the parameters and shows possible forms of it graphically.Figure 2
The failure rate function ofELk,m;δ,θ for some values of parameters.
(a)(b)Proposition 1.
The MRL function ofX∼ELk,m;δ,θ is(7)mx=1θΓkΓθx,m+1+δΓθx,k+1ΓkΓθx,m+δΓθx,k−x.Proof.
The MRL can be expressed by(8)mx=∫x∞RtdtRx.
The numerator of (8) can be simplified as(9)∫x∞Rtdt=1Γm+δ∫x∞Γθt,m+δΓkΓθt,kdt.
However,(10)∫x∞Γθt,kdt=∫x∞∫θt∞yk−1e−ydydt=∫θx∞∫xyθyk−1e−ydtdy=∫θx∞yk+1−1e−ydy−x∫θx∞yk−1e−ydy=1θΓθx,k+1−xΓθx,k.
Similarly,(11)∫x∞Γθt,mdt=1θΓθx,m+1−xΓθx,m.
Then, by (8)–(11), the result follows.
Theα-QRL is defined to be(12)qαx=infy:Ry+x=α¯Rx,x≥0,or more abbreviate qαx=R−1α¯Rx−x. When X∼ELk,m;δ,θ, the R−1 has not closed form and should be calculated numerically. For α=0.5, it is referred to median residual life which is an alternative for MRL in the reliability and survival analysis literature.
In Figure3, left side, we use (7) to draw the MRL. Also, in the right side of Figure 3, the median residual life has been plotted for some parameters.
Since the failure rate function may be increasing, bathtub-shaped, or increasing, then bathtub-shaped, the MRL function, and theα-QRL function may be decreasing, upside down bathtub-shaped, or decreasing then upside down bathtub-shaped, respectively (refer to Lai and Xie [25]). Figure 3 confirms this result graphically.
The point which maximizes the MRL/median residual life function is known as the burn-in time and has been attracted interest of many authors (see Mi [26]).Figure 3
(a) The MRL ofELk,m;δ,θ for some values of parameters. (b) The median residual life of ELk,m;δ,θ for some parameter values.
(a)(b)


## 2.1. Dynamic Measures

One of the most elementary measures in reliability theory and survival analysis is the reliability function,Rx=1−Fx. Moreover, dynamic measures like the failure rate function, the mean residual life (MRL) function, and the α-quantile residual life (α-QRL) function play key role in investigation the behaviour of lifetime. Let X∼ELk,m;δ,θ, then the failure rate function is(6)hx=fxRx=Γkθxm−1+δθxk−1ΓkΓθx,m+δΓθx,kθe−θx.Applying Glaser’s technique, this failure rate function may exhibit increasing, bathtub-shaped, or increasing then bathtub-shaped forms (see Glaser [24] or Theorem 2.8 in the study by Lai and Xie [25]). Figure 2 draws the failure rate function for some values of the parameters and shows possible forms of it graphically.Figure 2
The failure rate function ofELk,m;δ,θ for some values of parameters.
(a)(b)Proposition 1.
The MRL function ofX∼ELk,m;δ,θ is(7)mx=1θΓkΓθx,m+1+δΓθx,k+1ΓkΓθx,m+δΓθx,k−x.Proof.
The MRL can be expressed by(8)mx=∫x∞RtdtRx.
The numerator of (8) can be simplified as(9)∫x∞Rtdt=1Γm+δ∫x∞Γθt,m+δΓkΓθt,kdt.
However,(10)∫x∞Γθt,kdt=∫x∞∫θt∞yk−1e−ydydt=∫θx∞∫xyθyk−1e−ydtdy=∫θx∞yk+1−1e−ydy−x∫θx∞yk−1e−ydy=1θΓθx,k+1−xΓθx,k.
Similarly,(11)∫x∞Γθt,mdt=1θΓθx,m+1−xΓθx,m.
Then, by (8)–(11), the result follows.
Theα-QRL is defined to be(12)qαx=infy:Ry+x=α¯Rx,x≥0,or more abbreviate qαx=R−1α¯Rx−x. When X∼ELk,m;δ,θ, the R−1 has not closed form and should be calculated numerically. For α=0.5, it is referred to median residual life which is an alternative for MRL in the reliability and survival analysis literature.
In Figure3, left side, we use (7) to draw the MRL. Also, in the right side of Figure 3, the median residual life has been plotted for some parameters.
Since the failure rate function may be increasing, bathtub-shaped, or increasing, then bathtub-shaped, the MRL function, and theα-QRL function may be decreasing, upside down bathtub-shaped, or decreasing then upside down bathtub-shaped, respectively (refer to Lai and Xie [25]). Figure 3 confirms this result graphically.
The point which maximizes the MRL/median residual life function is known as the burn-in time and has been attracted interest of many authors (see Mi [26]).Figure 3
(a) The MRL ofELk,m;δ,θ for some values of parameters. (b) The median residual life of ELk,m;δ,θ for some parameter values.
(a)(b)


## 3. Power Extended Lindley

Now, we introduce the power Lindley extended (PEL) distribution with parametersk>0, m>0, k≠m, δ>0, θ>0, and γ>0, namely, PELk,m;δ,θ,γ, with the PDF as follows:(13)fx=θγxγ−1Γm+δδθxγk−1Γk+θxγm−1e−θxγ,x≥0.Form=2, k=1, θ=δ, and γ=1, it is the Lindley model. On the other hand, when δ=0 and γ=1, it reduces to the gamma distribution with PDF as follows:(14)fx=θmΓmxm−1e−θx,x>0.IfY∼ELk,m;δ,θ, then X=Y1/γ∼PELk,m;δ,θ,γ. This statement is useful for simulating random samples of PELk,m;δ,θ,γ and studying its properties.Figure4 shows the PDF for some various values of the parameters and indicates that the PDF accommodates decreasing, unimodal, and biomodal forms.Figure 4
The PDF ofPELk,m;δ,θ,γ for some values of the parameters.
(a)(b)It is straightforward to show that the CDF ofPELk,m;δ,θ,γ is(15)Fx=1−1Γm+δδΓkΓθxγ,k+Γθxγ,m,x≥0,in which Γu,α=∫u∞tα−1e−tdt is the upper incomplete gamma function. The rth moment of X∼PELk,m;δ,θ,γ is(16)EXr=θ−r/γ1Γm+δδΓkΓk+rγ+Γm+rγ.It is clear that whenk>1 and m>1, EXr>θ−r/γ. By the fact that etX=∑j=0∞tXj/j! and applying (16), the moment-generating function can be simplified to(17)EetX=∑j=0∞tjj!θ−j/γ1Γm+δδΓkΓk+jγ+Γm+jγ.


### 3.1. Dynamic Measures

LetX∼PELk,m;δ,θ,γ with the reliability function Rx, then the failure rate function is(18)hx=fxRx=δθxγk−1+Γkθxγm−1δΓθxγ,k+ΓkΓθxγ,mθγxγ−1e−θxγ.This failure rate function will accommodate increasing, bathtub, and increasing then bathtub-shaped forms since it is a power model of the baseline modelELk,m;δ,θ. The form of the failure rate of the power model depends on the baseline model and the value of γ. For example, if the failure rate function of the baseline model be increasing and γ>1, then the failure rate function of the power model will be clearly increasing. If the failure rate function of the baseline model shows bathtub shape and γ>1, then the failure rate function of the power model will be bathtub-shaped or increasing and so on.Figure5 draws the failure rate function for some values of parameters and confirms that it accommodates increasing, bathtub, and increasing then bathtub-shaped forms graphically.Figure 5
The failure rate function ofPELk,m;δ,θ,γ for some values of the parameters.
(a)(b)Proposition 2.
The MRL function ofX∼PELk,m;δ,θ,γ is(19)mx=θ−1/γδΓθxγ,k+1/γ+ΓkΓθxγ,m+1/γδΓθxγ,k+ΓkΓθxγ,m−x.Proof.
The MRL can be expressed by(20)mx=∫x∞RtdtRx.
The numerator of (20) can be simplified as(21)∫x∞Rtdt=1Γm+δ∫x∞δΓkΓθtγ,k+Γθtγ,mdt.
However,(22)∫x∞Γθtγ,kdt=∫x∞∫θtγ∞yk−1e−ydydt=∫θxγ∞∫xy/θ1/γyk−1e−ydtdy=θ−1/γ∫θxγ∞yk+1/γ−1e−ydy−x∫θxγ∞yk−1e−ydy=θ−1/γΓθxγ,k+1γ−xΓθxγ,k.
Similarly,(23)∫x∞Γθtγ,mdt=θ−1/γΓθxγ,m+1γ−xΓθxγ,m.
Then, by (20)–(23), the result follows.
Theα-QRL is defined to be(24)qαx=infy:Ry+x=α¯Rx,x≥0,or more abbreviated, qαx=R−1α¯Rx−x. When X∼PELk,m;δ,θ,γ, the R−1 has not closed form and should be calculated numerically.
Similarly, since the failure rate function shows increasing, bathtub-shaped, or increasing then bathtub-shaped, the MRL function and theα-QRL function may be decreasing, upside down bathtub-shaped, or decreasing then upside down bathtub-shaped, respectively (see Lai and Xie [25]). Figures 6 and 7 show the MRL and the median residual life functions for some parameters and confirm the mentioned forms graphically.Figure 6
Some forms of the MRL ofPELk,m;δ,θ,γ.
(a)(b)Figure 7
Some forms of the median residual life function ofPELk,m;δ,θ,γ.
(a)(b)


## 3.1. Dynamic Measures

LetX∼PELk,m;δ,θ,γ with the reliability function Rx, then the failure rate function is(18)hx=fxRx=δθxγk−1+Γkθxγm−1δΓθxγ,k+ΓkΓθxγ,mθγxγ−1e−θxγ.This failure rate function will accommodate increasing, bathtub, and increasing then bathtub-shaped forms since it is a power model of the baseline modelELk,m;δ,θ. The form of the failure rate of the power model depends on the baseline model and the value of γ. For example, if the failure rate function of the baseline model be increasing and γ>1, then the failure rate function of the power model will be clearly increasing. If the failure rate function of the baseline model shows bathtub shape and γ>1, then the failure rate function of the power model will be bathtub-shaped or increasing and so on.Figure5 draws the failure rate function for some values of parameters and confirms that it accommodates increasing, bathtub, and increasing then bathtub-shaped forms graphically.Figure 5
The failure rate function ofPELk,m;δ,θ,γ for some values of the parameters.
(a)(b)Proposition 2.
The MRL function ofX∼PELk,m;δ,θ,γ is(19)mx=θ−1/γδΓθxγ,k+1/γ+ΓkΓθxγ,m+1/γδΓθxγ,k+ΓkΓθxγ,m−x.Proof.
The MRL can be expressed by(20)mx=∫x∞RtdtRx.
The numerator of (20) can be simplified as(21)∫x∞Rtdt=1Γm+δ∫x∞δΓkΓθtγ,k+Γθtγ,mdt.
However,(22)∫x∞Γθtγ,kdt=∫x∞∫θtγ∞yk−1e−ydydt=∫θxγ∞∫xy/θ1/γyk−1e−ydtdy=θ−1/γ∫θxγ∞yk+1/γ−1e−ydy−x∫θxγ∞yk−1e−ydy=θ−1/γΓθxγ,k+1γ−xΓθxγ,k.
Similarly,(23)∫x∞Γθtγ,mdt=θ−1/γΓθxγ,m+1γ−xΓθxγ,m.
Then, by (20)–(23), the result follows.
Theα-QRL is defined to be(24)qαx=infy:Ry+x=α¯Rx,x≥0,or more abbreviated, qαx=R−1α¯Rx−x. When X∼PELk,m;δ,θ,γ, the R−1 has not closed form and should be calculated numerically.
Similarly, since the failure rate function shows increasing, bathtub-shaped, or increasing then bathtub-shaped, the MRL function and theα-QRL function may be decreasing, upside down bathtub-shaped, or decreasing then upside down bathtub-shaped, respectively (see Lai and Xie [25]). Figures 6 and 7 show the MRL and the median residual life functions for some parameters and confirm the mentioned forms graphically.Figure 6
Some forms of the MRL ofPELk,m;δ,θ,γ.
(a)(b)Figure 7
Some forms of the median residual life function ofPELk,m;δ,θ,γ.
(a)(b)


## 4. Parameter Estimation

LetXi, i=1,2,…,n represent an independent and identically distributed (iid) sample from ELk,m;δ,θ, and k and m are known. Via the moment method, by (5), δ,θ can be estimated by solving the following equations in terms of them:(25)X¯=1θ1Γm+δΓm+1+kδ,X2¯=1θ21Γm+δΓm+2+kk+1δ.Form≥1 and k≥1, the right side of these equations is greater than 1/θ and 1/θ2, respectively. So, one lower bound for θ is max1/X¯,1/X2¯0.5.The log-likelihood function forδ,θ when k and m are known is(26)lδ,θ;X=nlnθ−nlnΓm+δ+∑i=1nlnδΓkθXik−1+θXim−1−θ∑i=1nXi.The log-likelihood equations are(27)∂lδ,θ;X∂δ=−nΓm+δ+∑i=1nθXik−1ΓkθXim−1+δθXik−1=0,(28)∂lδ,θ;X∂θ=nθ+∑i=1nΓkm−1XiθXim−2+δk−1XiθXik−2ΓkθXim−1+δθXik−1−∑i=1nXi=0.The MLE can be computed by directly maximizing (26) or solving the log-likelihood equations (27) and (28) simultaneously.The Fisher information matrix in one sample of sizen about δ,θ is(29)K=E−∂2l∂δ2E−∂2l∂δ∂θE−∂2l∂θ∂δE−∂2l∂θ2,where l=lnfX. Let K−1 be the inverse of the information matrix, then Varδ^,θ^≈n−1K−1. Moreover, if Xi, i=1,2,…,n represents an iid sample from ELk,m;δ0,θ0, then(30)nδ^−δ0,θ^−θ0T∼N0,K−1,asymptotically, which can be applied for constructing confidence intervals for the parameters or in hypothesis testing.Whenk and/or m are unknown which is usually the case, we can estimate δ,θ for a suitable set of values of k and m and compare the estimated distribution to the empirical distribution by applying statistics like Kolmogorov–Smirnov (K-S) statistic.


### 4.1. Parameter Estimation of the Power Extended Lindley

LetXi, i=1,2,…,n represent an iid sample from PELk,m;δ,θ,γ, and k and m are known. Via the moment method, by (16), the triple δ,θ,γ can be estimated by finding their solution in the following system of equations:(31)X¯=θ−1/γ1Γm+δδΓkΓk+1γ+Γm+1γ,X2¯=θ−2/γ1Γm+δδΓkΓk+2γ+Γm+2γ,X3¯=θ−3/γ1Γm+δδΓkΓk+3γ+Γm+3γ.The log-likelihood function forδ,θ,γ when k and m are known is(32)lδ,θ,γ;X=nlnγ+γ−1∑i=1nlnXi+nlnθ−nlnΓm+δ+∑i=1nlnδΓkθXiγk−1+θXiγm−1−θ∑i=1nXiγ.The log-likelihood equations are(33)∂lδ,θ,γ;X∂δ=−nΓm+δ+∑i=1nθXiγk−1ΓkθXiγm−1+δθXiγk−1=0,(34)∂lδ,θ,γ;X∂θ=nθ+∑i=1nΓkm−1XiγθXiγm−2+δk−1XiγθXiγk−2ΓkθXiγm−1+δθXiγk−1−∑i=1nXiγ=0,(35)∂lδ,θ,γ;X∂γ=nγ+∑i=1nlnXi+∑i=1nlnXiδk−1θXiγk−1+Γkm−1θXiγm−1δθXiγk−1+ΓkθXiγm−1−θ∑i=1nXiγlnXi=0.The MLE can be computed by maximizing (32) directly or solving the log-likelihood equations (33)–(35) simultaneously.The Fisher information matrix in one sample of sizen about δ,θ,γ is(36)K=E−∂2l∂δ2E−∂2l∂δ∂θE−∂2l∂δ∂γE−∂2l∂θ∂δE−∂2l∂θ2E−∂2l∂θ∂γE−∂2l∂γ∂δE−∂2l∂γ∂θE−∂2l∂γ2,where l=lnfX. Assume that K−1 be the inverse of the information matrix, then Varδ^,θ^,γ^≈n−1K−1. Furthermore, let Xi, i=1,2,…,n represent an iid sample from PELk,m;δ0,θ0,γ0, then(37)nδ^−δ0,θ^−θ0,γ^−γ0T∼N0,K−1,asymptotically, which can be applied for constructing confidence intervals for the parameters or in hypothesis testing.Similarly, whenk and/or m are unknown, we can estimate δ,θ,γ for a suitable set of values of k and m and compare the estimated distribution to the empirical distribution by K-S statistic or other similar statistics.


### 4.2. Right-Censored Data

Assume that we have one iid random sampleXi, i=1,2,…,n from the EL (PEL) which is exposed to right censorship. Then, Xi is censored from right by a censoring random variable Ci, if Ci≤Xi, so the only information about event time is that it is greater than censoring time Ci. Thus, the observations consist of Ti=minXi,Ci and di, where di=1, when the event is not censored, Xi≤Ci, and di=0, when the event is censored, Xi>Ci. With the censored data in hand, the log-likelihood function is denoted by lcδ,θ;t,dlcδ,θ,γ;t,d and equals to(38)∑i=1ndilnfti+∑i=1n1−dilnRti,where f and R show the density and the reliability functions of the EL (PEL), respectively. We can maximize this function in terms of δ,θδ,θ,γ to find the MLE and estimate its variance by reverse of the Fisher information matrix.


## 4.1. Parameter Estimation of the Power Extended Lindley

LetXi, i=1,2,…,n represent an iid sample from PELk,m;δ,θ,γ, and k and m are known. Via the moment method, by (16), the triple δ,θ,γ can be estimated by finding their solution in the following system of equations:(31)X¯=θ−1/γ1Γm+δδΓkΓk+1γ+Γm+1γ,X2¯=θ−2/γ1Γm+δδΓkΓk+2γ+Γm+2γ,X3¯=θ−3/γ1Γm+δδΓkΓk+3γ+Γm+3γ.The log-likelihood function forδ,θ,γ when k and m are known is(32)lδ,θ,γ;X=nlnγ+γ−1∑i=1nlnXi+nlnθ−nlnΓm+δ+∑i=1nlnδΓkθXiγk−1+θXiγm−1−θ∑i=1nXiγ.The log-likelihood equations are(33)∂lδ,θ,γ;X∂δ=−nΓm+δ+∑i=1nθXiγk−1ΓkθXiγm−1+δθXiγk−1=0,(34)∂lδ,θ,γ;X∂θ=nθ+∑i=1nΓkm−1XiγθXiγm−2+δk−1XiγθXiγk−2ΓkθXiγm−1+δθXiγk−1−∑i=1nXiγ=0,(35)∂lδ,θ,γ;X∂γ=nγ+∑i=1nlnXi+∑i=1nlnXiδk−1θXiγk−1+Γkm−1θXiγm−1δθXiγk−1+ΓkθXiγm−1−θ∑i=1nXiγlnXi=0.The MLE can be computed by maximizing (32) directly or solving the log-likelihood equations (33)–(35) simultaneously.The Fisher information matrix in one sample of sizen about δ,θ,γ is(36)K=E−∂2l∂δ2E−∂2l∂δ∂θE−∂2l∂δ∂γE−∂2l∂θ∂δE−∂2l∂θ2E−∂2l∂θ∂γE−∂2l∂γ∂δE−∂2l∂γ∂θE−∂2l∂γ2,where l=lnfX. Assume that K−1 be the inverse of the information matrix, then Varδ^,θ^,γ^≈n−1K−1. Furthermore, let Xi, i=1,2,…,n represent an iid sample from PELk,m;δ0,θ0,γ0, then(37)nδ^−δ0,θ^−θ0,γ^−γ0T∼N0,K−1,asymptotically, which can be applied for constructing confidence intervals for the parameters or in hypothesis testing.Similarly, whenk and/or m are unknown, we can estimate δ,θ,γ for a suitable set of values of k and m and compare the estimated distribution to the empirical distribution by K-S statistic or other similar statistics.


## 4.2. Right-Censored Data

Assume that we have one iid random sampleXi, i=1,2,…,n from the EL (PEL) which is exposed to right censorship. Then, Xi is censored from right by a censoring random variable Ci, if Ci≤Xi, so the only information about event time is that it is greater than censoring time Ci. Thus, the observations consist of Ti=minXi,Ci and di, where di=1, when the event is not censored, Xi≤Ci, and di=0, when the event is censored, Xi>Ci. With the censored data in hand, the log-likelihood function is denoted by lcδ,θ;t,dlcδ,θ,γ;t,d and equals to(38)∑i=1ndilnfti+∑i=1n1−dilnRti,where f and R show the density and the reliability functions of the EL (PEL), respectively. We can maximize this function in terms of δ,θδ,θ,γ to find the MLE and estimate its variance by reverse of the Fisher information matrix.


## 5. Simulation Study

The proposed distribution is a mixture of two gamma distribution, so we can follow the below steps to generate random samples of sizen of it.(1)
Simulate one random instance from binomial distribution with parametersn and δ/Γm+δ. Let n1 be the generated instance and n2=n−n1.(2)
Simulate one random sample of sizen1 from Gk,θ and one random sample of size n2 from Gm,θ. Then, merge these two sample to provide one random sample from ELk,m,δ,θ.(3)
To simulate one sample of sizen from PELk,m;δ,θ,γ, we simulate one sample from ELk,m,δ,θ as described in the previous steps. Let Y1,Y2,…,Yn show the simulated sample from ELk,m,δ,θ. Then, we raise this sample to power 1/γ, i.e., Y11/γ,Y21/γ,…,Yn1/γ.(4)
In the case that the sample must be censored from right, we assume thatCi follows the uniform distribution on the interval 0,M. Given the censorship rate, p and M can be computed through solving the following equation:(39)∫0MRtdt=Mp,where R is the reliability function of the extended Lindley.The result of this investigation for the EL model is abstracted in Table1. For every cell of this table, r=500 replicates have been simulated. For each replication, the MLE of δ,θ, δ^,θ^, has been computed. Then, three pairs Bδ,Bθ, ABδ,ABθ, and MSEδ,MSEθ in which(40)Bδ=1r∑i=1rδ^i−δ,ABδ=1r∑i=1rδ^i−δ,MSEδ=1r∑i=1rδ^i−δ2,and Bθ, ABθ, and MSEθ are defined similarly have been calculated. Results of the simulations in Table 1 indicate that MLE of δ,θ is efficient and consistent. Moreover, the following observations are notable:(i)
All measures have smaller values for larger sample size. The MLE has a small positive bias for uncensored data which moves toward negative values for censored samples.(ii)
The measures show larger values for largerθs (δs).(iii)
For censored samples,ABθ and MSEθ are larger.Table2 contains the results of simulations for the PEL distribution. k and m have been considered to be 4 and 1, respectively. For every cell of this table, r=500 replicates have been simulated, and for each replication, the MLE of δ,θ,γ and δ^,θ^,γ^ has been calculated. Then, the triples Bδ,Bθ,Bγ, ABδ,ABθ,ABγ, and MSEδ,MSEθ,MSEγ have been calculated. The simulation results indicate that the MLE of δ,θ,γ is efficient and consistent. Moreover, the following observations are notable.(iv)
The MSE and absolute error have smaller values for larger sample size.(v)
For censored cases, the MSE and absolute error show greater values.(vi)•
The measures aboutδ show larger values for larger δ. Similar result holds for θ and γ.Table 1
Simulation results for MLE ofθ from ELk,m,δ,θ. Every cell consists of pairs Bδ,Bθ, ABδ,ABθ, and MSEδ,MSEθ from top to bottom.
nk,mp=0p=0.20δ=2,θ=0.01δ=1,θ=2δ=2,θ=0.01δ=1,θ=2100(3,1)(0.564546, 0.000322)(0.125512, 0.002711)(0.201321, −0.000122)(−0.630433, −0.669626)(0.862838, 0.000784)(0.406587, 0.207322)(0.824443, 0.001039)(0.795519, 0.687137)(1.237973, 0.000001)(0.311982, 0.076444)(1.428182, 0.000002)(0.773555, 0.606325)(4,2)(0.668760, 0.000149)(0.251394, −0.002273)(0.650087, −0.000270)(0.371670, −.027948)(1.145145, 0.000577)(0.666065, 0.224162)(1.559371, 0.001144)(0.886215, 0.278360)(1.850450, 0.0000006)(0.843101, 0.081450)(7.354127, 0.000002)(2.729726, 0.116106)150(3,1)(0.365396, 0.000155)(0.085817, 0.008477)(0.128090, −0.000109)(−0.528556, −0.556155)(0.665277, 0.000581)(0.327193, 0.171947)(0.612812, 0.000792)(0.655863, 0.594525)(0.814574, 0.0000005)(0.184370, 0.051351)(0.740215, 0.000001)(0.572858, 0.520272)(4,2)(0.592613, 0.000150)(0.147898, −0.017771)(0.276481, −0.000337)(0.107956, −0.043725)(1.003509, 0.000489)(0.551938, 0.187519)(1.147521, 0.000996)(0.609155, 0.227483)(1.519897, 0.0000004)(0.543016, 0.057536)(3.144793, 0.000002)(0.695320, 0.082204)Table 2
Simulation results for MLE of the parameters ofPELk,m,δ,θ,γ. Every cell consists of triples Bδ,Bθ,Bγ, ABδ,ABθ,ABγ, and MSEδ,MSEθ,MSEγ from top to bottom.
pnγδ=3,θ=0.05δ=2,θ=0.0101001.2(−0.102169, −0.007042, 0.122675)(−0.077950, 0.000187, 0.085795)(1.385808, 0.026254, 0.189938)(0.799550, 0.007379, 0.169717)(3.532168, 0.001054, 0.074171)(1.004847, 0.000099, 0.045673)0.8(−0.510318, −0.011244, 0.014736)(−0.463949, −0.002733, 0.125280)(1.310887, 0.025964, 0.141285)(0.902582, 0.007098, 0.158792)(3.075335, 0.000937, 0.041458)(1.187285, 0.000078, 0.041090)1501.2(0.033280, −0.003561, 0.072078)(−0.066741, 0.000316, 0.055330)(1.124049, 0.021518, 0.137609)(0.658367, 0.006076, 0.130425)(3.265883, 0.000760, 0.041036)(0.721868, 0.000070, 0.029051)0.8(−0.186257, −0.005086, 0.065326)(−0.569158, −0.003036, 0.115042)(1.131052, 0.022653, 0.109088)(0.832820, 0.006230, 0.142214)(2.633615, 0.000779, 0.027734)(1.004401, 0.000059, 0.035583)0.21001.2(−0.158072, −0.003071, 0.125427)(−0.136315, 0.002267, 0.092781)(1.412975, 0.030999, 0.218746)(0.899895, 0.009618, 0.210720)(3.570973, 0.001677, 0.099605)(1.453491, 0.000222, 0.074069)0.8(−0.331269, −0.005152, 0.100935)(−0.711753, −0.004670, 0.195593)(1.482429, 0.032797, 0.158092)(0.996478, 0.009042, 0.223078)(4.623204, 0.002162, 0.050992)(1.371019, 0.000156, 0.060592)1501.2(−0.092884, 0.000442, 0.082341)(−0.124146, 0.001331, 0.076157)(1.194005, 0.029246, 0.184984)(0.783033, 0.007909, 0.172482)(2.407211, 0.001456, 0.067009)(1.033839, 0.000156, 0.054197)0.8(−0.006219, −0.000185, 0.049270)(−0.672792, −0.005171, 0.193702)(1.220938, 0.271811, 0.112398)(0.886939, 0.008259, 0.218931)(2.825691, 0.001249, 0.026412)(1.009560, 0.000079, 0.058949)


## 6. Applications

In this section, four data sets, (see Tables3–6) have been considered. For every data set, the ELk,m;δ,θ and PELk,m;δ,θ,γ have been fitted, for all pairs k,m where k=1,2,…,10 and m=1,2,…,10. Then, based on the K-S statistic and AIC, the best values of k and m are selected. In a comparative analysis, the quasi Lindley (QL) and the power quasi Lindley (PQL) distributions have been fitted to these data sets too. The K-S, the Cramér–von Mises (CVM), and the Anderson–Darling (AD) statistics along with their corresponding p values have been computed. Also, the Akaike information criterion (AIC) has been reported.Table 3
Number of cycles to failure.
1520384261768698121146149157175176180180198220224251264282321325653—Table 4
Waiting times (in minutes) of customers to receive service in a bank.
0.80.81.31.51.81.91.92.12.62.72.93.13.23.33.53.64.04.14.24.24.34.34.44.44.64.74.74.84.94.95.05.35.55.75.76.16.26.26.26.36.76.97.17.17.17.17.47.67.78.08.28.68.68.68.88.88.98.99.59.69.79.810.710.911.011.011.111.211.211.511.912.412.512.913.013.113.313.613.713.914.115.415.417.317.318.118.218.418.919.019.920.621.321.421.923.027.031.633.138.5Table 5
Number of cycles before failure of 60 electrical appliances, reported by Lawless [27].
001400340059006100690080012301420165021003810464047905560574083909170969099110641088109111741270127513551397147715781649170218931932200121612292232623372628278528112886299331223248371537903857391241004106411643154510458452675299558360659701—————Table 6
Survival times of a group of patients suffering from head and neck cancer.
12.2023.5623.7425.8731.983741.3547.3855.4658.3663.4768.4678.2674.4781.438492941101121191271301331401461551591731791941952092492813193394324695196337258171776————————Table3 shows the number of cycles to failure for 25 specimens of yarn, considered in Shanker [28]. The results of fit are gathered in Table 7. Clearly, both EL and PEL describe data very well and in a very close competition. Among the compared models, the PEL gives the best fit in terms of K-S, CVM, and AD statistics. However, the AIC of QL is smaller than others. Figure 8(a) shows the empirical CDF and fitted CDF of EL and PEL and graphically confirms a good fit. Figure 9(a), shows the PDF of the fitted EL and PEL distributions. Also, left side of Figure 10 shows the scaled total time on test (TTT) plot of the estimated EL and PEL distributions and reveals an increasing failure rate model.Table 7
Results of fitting models.
Data setModelkmδ^,θ^,γ^AICK-S statisticp valueCVM statisticp valueAD statisticp valueTable3EL14(1.8070, 0.0185)309.43340.0876, 0.99060.0305, 0.97660.2805, 0.9517PEL14(1.5846, 0.0235, 0.9598)311.41350.0822, 0.99590.0302, 0.97750.2714, 0.9576QL——(0.0843, 0.0107)308.91540.1323, 0.77360.0542, 0.85470.3327, 0.9109PQL——(2.5383, 0.0010, 1.3536)310.96130.1101, 0.92230.0474, 0.89590.3269, 0.9159Table4EL72(0.0051, 0.2051)638.58090.0379, 0.99870.0236, 0.99240.1614, 0.9976PEL96(0.000001, 1.7927, 0.5545)640.23480.0356, 0.99960.0136, 0.99890.1382, 0.9993QL——(0.3×10−6, 0.2024)638.60140.0422, 0.99410.0282, 0.98190.1836, 0.9943PQL——(480.295, 0.0304, 1.4587)643.46150.0590, 0.87660.0638, 0.79060.4150, 0.8333Table5EL41(0.7893, 0.001)1045.15900.0751, 0.86200.0608, 0.81060.5470, 0.6987PEL14(2.6871, 0.0071, 0.7976)1043.48500.0642, 0.95180.0271, 0.98540.2137, 0.9862QL——(2.4425, 0.00058)1046.510.0830, 0.77170.0787, 0.70100.7385, 0.5269PQL——(2.8669, 0.0007, 0.9755)1048.5250.0769, 0.84300.0892, 0.64200.7207, 0.5411Table6EL102(0.1451, 0.0135)565.13870.0930, 0.80770.1054, 0.56060.6927, 0.5640PEL102(0.0952, 0.0326, 0.8310)562.96480.0571, 0.99710.0172, 0.99900.1515, 0.9985QL——(761.09, 0.0044)568.01930.1454, 0.28160.1777, 0.31590.9673, 0.3744PQL——(565.2588, 0.00637, 0.9407)569.68340.1310, 0.40230.1438, 0.41000.8765, 0.4283Figure 8
(a) The empirical and fitted CDF of data in Table3. (b) The empirical and fitted CDF of data in Table 4.
(a)(b)Figure 9
The estimated PDF of the EL and the PEL of data sets of Table3 (a) and Table 4 (b), respectively.
(a)(b)Figure 10
The estimated scaled TTT plot of the EL and the PEL of data sets of Table3 (a) and Table 4 (b), respectively.
(a)(b)The second data set (see Table4) consists of 100 waiting times of customers of a bank reported by Shanker [28]. Similarly, Table 7 shows the results of fit for this example. The PEL gives the lowest value for the K-S, CVM, and AD statistics. So, PEL outperforms other models from this point of view. The empirical and fitted CDF for EL and PEL has been plotted in Figure 8(b) and confirms a good fit. Also, the estimated PDF and the scaled TTT plot have been drawn in Figures 9 and 10, respectively.Table5 shows the cycles to failure for 60 electrical appliances analyzed by Lawless [27]. It can be seen from Table 7 that the PEL gives a better fit than other rivals due to the lower AIC and K-S, CVM, and AD statistics. Figure 11(a) depicts the fit graphically. The estimated PDF and the scaled TTT plot have been plotted in Figures 12 and 13, respectively.Figure 11
(a) The empirical and fitted CDF of data in Table5. (b) The empirical and fitted CDF of data in Table 6.
(a)(b)Figure 12
The estimated PDF of the EL and the PEL of data sets of Table5 (a) and Table 6 (b), respectively.
(a)(b)Figure 13
The estimated scaled TTT plot of the EL and the PEL of data sets of Table5 (a) and Table 6 (b), respectively.
(a)(b)Table6 shows survival times of a group of patients suffering from head and neck cancer treated by a combination of radiotherapy and chemotherapy, reported by Efron [29]. Again the PEL distribution gives the best fit among compared models. Figure 11(b) shows the empirical and fitted CDF and indicates a good fit for both models apparently. Moreover, Figures 12 and 13 show the estimated PDF and the scaled TTT plots of the EL and PEL models.


## 7. Conclusions

Lindley distribution and its generalizations are useful in reliability theory and for analyzing real data sets. Here, we introduce one new scale-invariant generalization of the Lindley distribution and its power generalization and study their traits and the problem of estimation of their parameters. The models are not very complicated but accommodate different and useful forms of hazard rate function (MRL or median residual life functions). The simulation results show that MLE is applicable and efficient. Four real data sets have been analyzed, and the results show that the proposed models can fit them conveniently.



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