A new generalized linear exponential distribution and its applications

Acta Mathematicae Applicatae Sinica, English Series

Vol. 30, No. 4 (2014) 1049–1062

DOI: 10.1007/s10255-014-0442-4http://www.ApplMath.com.cn & www.SpringerLink.com

Acta Mathema�cae Applicatae Sinica,English Series© The Editorial Office of AMAS & Springer-Verlag Berlin Heidelberg 2014

A New Generalized Linear Exponential Distribution andIts ApplicationsYu-zhu TIAN1,3,†, Mao-zai TIAN2,3, Qian-qian ZHU3

1School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China

(E-mail: [email protected])2School of Statistics, Lanzhou University of Finance and Economics, Lanzhou 730020, China

(E-mail: [email protected])3Center for Applied Statistics, School of Statistics, Renmin University of China, Beijing 100872, China

Abstract A new generalized linear exponential distribution (NGLED) is considered in this paper which can be

deemed as a new and more flexible extension of linear exponential distribution. Some statistical properties for the

NGLED such as the hazard rate function, moments, quantiles are given. The maximum likelihood estimations

(MLE) of unknown parameters are also discussed. A simulation study and two real data analyzes are carried out

to illustrate that the new distribution is more flexible and effective than other popular distributions in modeling

lifetime data.

Keywords generalized linear exponential distribution (GLED); linear exponential distribution; Hazard rate

function; maximum likelihood estimation (MLE)

2000 MR Subject Classification O211.3

Introduction

Some distributions such as the exponential, Rayleigh, Weibull, linear exponential distributionare often used to model lifetime data. These distributions have several desirable properties andsatisfying interpretations which enable them to be used frequently in analyzing lifetime data.Especially, the linear exponential distribution, having exponential and Rayleigh distributionsas special cases, is a very useful distribution for modeling lifetime data with linearly increasingfailure rates. However, the linear exponential distribution does not provide a rational parametricfit for modeling phenomenon with decreasing, non linearly increasing, or non-monotone failurerates. Therefore, a lot of work is devoted to generalizing the linear exponential distributionfrom different viewpoints and propose some new and more flexible distributions. For example,Gupta and Kundu[1] studied the generalized exponential distribution. The generalized Rayleighdistribution was proposed by Surles and Padgett[12]. Lin et al.[6] discussed Bayesian inferenceof the linear hazard rate distribution. Sarhan and Zaindiu [15] proposed a modified Weibulldistribution. Sarhan and Kundu [14] studied the generalized linear failure rate distribution.Mahmoud and Alam[8] proposed the generalized linear exponential distribution. Zaindin andSarhan[16] proposed a new generalized Weibull distribution. Sarhan et al.[13] considered anexponentiated generalized linear exponential distribution which is also an extension of the

Manuscript received May 7, 2012. Revised March 26, 2013.Partially supported by National Natural Science Foundation of China (No.11271368), Beijing Philosophy andSocial Science Foundation Grant (No.12JGB051), Project of Ministry of Education supported by the SpecializedResearch Fund for the Doctoral Program of Higher Education of China (Grant No. 20130004110007), The KeyProgram of National Philosophy and Social Science Foundation Grant (No. 13AZD064), and China StatisticalResearch Project (No. 2011LZ031).†Corresponding author.

1050 Y.Z. TIAN, M.Z. TIAN, Q.Q. ZHU

linear exponential distribution. In this paper we consider another new generalization of thelinear exponential distribution denoted as the NGLED from the viewpoint of linear exponentialdistribution, and discuss its statistical properties and some applications. The NGLED includesfour parameters, and has probability density function (pdf) and cumulative distribution function(cdf) respectively as follows:

f(x; α, λ, β, γ) = α(λ + βγxγ−1)e−(λx+βxγ)(1 − e−(λx+βxγ))α−1, x > 0,

F (x; α, λ, β, γ) = (1 − e−(λx+βxγ))α, x > 0.(1)

where α, γ > 0 are shape parameters and λ, β are scale parameters, respectively. This modelhas also been considered in literature, see for example Zaindin and Sarhan[16]. It has somedistributions as its special cases.

For β = 0, Model (1) can be deemed as a generalized exponential distribution with twoparameters (see Gupta and Kundu[1]).

For λ = 0, Model (1) can be regarded as an exponentiated Weibull distribution that wasproposed by Mudholkar and Srivastava [10].

For λ = 0, γ = 2, Model (1) can be deemed as a generalized Rayleigh distribution includingtwo parameters (see Kundu and Raqab [4]).

For α = 1, γ = 2, Model (1) can be seen as a linear exponential distribution.For γ = 2, Model (1) can be deemed as a generalized linear failure rate distribution proposed

in Sarhan and Kundu[14].For α = 1, Model (1) can be regarded as a modified Weibull distribution proposed by Sarhan

and Zaindin[15].It is interesting to observe that the cdf of the NGLED represents the cdf of the maximum

of a simple random sample of size α from a modified Weibull distribution proposed in Sarhanand Zaindin [15] when α is a positive integer. Therefore, in this case, the NGLED providesthe distribution function of a parallel system when each component has a modified Weibulldistribution lifetime. Furthermore, the NGLED, due to its flexibility in accommodating differentcases of the hazard rate function, seems to be a suitable distribution family that can be appliedwidely in fitting lifetime data.

The remainder of this paper has the following organization. In Section 2, we give somesimple statistical properties of the NGLED such as moments, quantiles and so on. In Section3, parameter estimations of the NGLED are discussed and a simulation study is implemented.In Section 4, two real data analyses are provided to illustrate the proposed distribution. In thelast section, we draw some conclusions about this paper.

2 Statistical Properties

2.1 Raw Moments

Theorem 1. The k-th raw moment of the NGLED, denoted as μk, is easily obtained by thefollowing form

μk =∞∑

n=0

∞∑

s=0

(−1)n+s

(α

n

)· βs · (n + 1)(1−γ)s−k−γ

λsγ+k · s!

·[(n + 1)γ−1Γ(sγ + k + 1) +

βγ · Γ(sγ + k + γ)λγ

].

Proof. See Appendix A. Similar proof can also refer to Zaindin and Sarhan[16]. �

A New Generalized Linear Exponential Distribution and Its Applications 1051

Remark1. It is noted that μk, k = 1, 2, · · ·, in practice, are not easy to compute dueto the double sum and the gamma function including in the right part of the above formula.However, we can approximate the gamma function Γ(α) using the Stirling’s formula[11] asfollows Γ(α) � √

2παα−1/2e−α.

2.2 The Moment Generating Function

Theorem 2. The moment generating function of the NGLED, denoted as M(t), is obtainedas follows.

M(t) =∞∑

n=0

∞∑

s=0

(−1)n+s

(α

n

)· βs(n + 1)s

s!

[ λ · Γ(sγ + 1)((n + 1)λ − t)sγ+1

+βγ · Γ(sγ + γ)

((n + 1)λ − t)sγ+γ

], t ≤ λ.

Proof. See Appendix B. �

Remark2. We can easily obtain the characteristic function by replacing t with it in M(t).The characteristic function may be a more convenient tool due to its existence at any values. Wecan also approximate the gamma function Γ(α) in M(t) using the previous Stirling’s formula.

2.3 Modes and Quantiles

If the non-negative random variable X follows the NGLED Model (1), the mode which isdefined as the maximal value of the probability density function, denoted by x0, can be obtainednumerically by solving the following nonlinear equation

[α(λ + βγxγ−10 )2 − βγ(γ − 1)xγ−2

0 ] · e−(λx0+βxγ0 ) = (λ + βγxγ−1

0 )2 − βγ(γ − 1)xγ−20 .

The p-th subquantile xp of the NGLED is given by the following nonlinear equation βxγp +

λxp + ln(1− p1α ) = 0. Since we cannot obtain the closed form of xp, the numerical method can

be employed to get the convergent solution of the above nonlinear equation.

Figure 1. The pdf and the hrf when α = 3, λ = 1.2, β = 1.5, γ = 1.


Figure 2. The pdf and the hrf when α = 1, λ = 0.8, β = 7, γ = 1.

2.4 The Survival Function and the Hazard Rate Function (hrf)

We can obtain the survival function of the NGLED Model (1) as follows.

R(x; α, λ, β, γ) = 1 − (1 − e−(λx+βxγ))α, x ≥ 0.

The hrf of the NGLED Model (1) is obtained as follows.

h(x) =f(x)R(x)

=α(λ + βγxγ−1)e−(λx+βxγ)(1 − e−(λx+βxγ))α−1

1 − (1 − e−(λx+βxγ))α, x ≥ 0.

The pdf and the hrf for different parameters are presented from Figure 1 to Figure 7.According to Figure 1 to Figure 7, we can see that the pdf of the NGLED can be decreasingor unimodal, and the hrf can be decreasing, increasing, a bathtub shaped function or constantand so on. Hence, the NGLED model can replace many popular distributions in analyzing thelifetime data due to the rich classes of its hrf.

Figure 3. The pdf and the hrf when α = 0.6, λ = 1.5, β = 0.6, γ = 1.8.

Figure 4. The pdf and the hrf when α = 5, λ = 0.6, β = 0.5, γ = 1.5.


2.5 Order Statistics

If X(1) ≤ X(2) ≤ · · · ≤ X(n) denote the order statistics of a random sample X1, X2, · · · , Xn

from a continuous population with the cdf F (x) and the pdf f(x), then the pdf of X(j) is givenby

fX(j)(x) =n!

(j − 1)!(n − j)!f(x)[F (x)]j−1 [1 − F (x)]n−j .

Hence, the pdf of the j-th order statistic for the NGLED (1) can be obtained by

fX(j)(x) =n!

(j − 1)!(n − j)!α(λ + βγxγ−1)e−(λx+βxγ)(1 − e−(λx+βxγ))α−1

· [(1 − e−(λx+βxγ))α]j−1[1 − (1 − e−(λx+βxγ))α]n−j

.

Therefore, the pdf of the largest order statistic X(n) is

fX(n)(x) = nα(λ + βγxγ−1)e−(λx+βxγ)(1 − e−(λx+βxγ))α−1[(1 − e−(λx+βxγ))α

]n−1.

And the pdf of the smallest order statistic X(1) is

fX(1)(x) = nα(λ + βγxγ−1)e−(λx+βxγ)(1 − e−(λx+βxγ))α−1 · [1 − (1 − e−(λx+βxγ))α]n−1.

Figure 5. The pdf and the hrf when α = 1, λ = 2, β = 0.3, γ = 4.

Figure 6. The pdf and the hrf when α = 1, λ0.5, β = 1, γ = 3.


Figure 7. The pdf and the hrf when α = 0.5, λ0.5, β = 1.5, γ = 1.

3 The MLEs

In this section, we consider the MLEs of all unknown parameters of the NGLED. Suppose anobserved random sample, denoted as x = (x1, x2, · · · , xn), with size n, which is the independentidentical distribution sample from the NGLED. Then, the likelihood function of the NGLEDis given by

L(α, λ, β, γ) =n∏

i=1

α(λ + βγxγ−1i )e−(λxi+βxγ

i)(1 − e−(λxi+βxγ

i))α−1

=αne−

n∑i=1

(λxi+βxγi) n∏

i=1

(λ + βγxγ−1i )(1 − e−(λxi+βxγ

i))α−1.

The log-likelihood function can be written as

ln L(α, λ, β, γ) =n · ln α −n∑

i=1

(λxi + βxγi ) +

n∑

i=1

ln(λ + βγxγ−1i )

+ (α − 1)n∑

i=1

ln(1 − e−(λxi+βxγi)).

Differentiating the above lnL(α, λ, β, γ) with respect to parameters α, λ, β and γ, respec-tively, and equating them to zero gives

∂ ln L(α, λ, β, γ)∂α

=n

α+

n∑

i=1

ln(1 − e−(λxi+βxγi)) = 0,

∂ ln L(α, λ, β, γ)∂λ

= −n∑

i=1

xi +n∑

i=1

1λ + βγxγ−1

i

+ (α − 1)n∑

i=1

xi

e(λxi+βxγi) − 1

= 0,

∂ ln L(α, λ, β, γ)∂β

= −n∑

i=1

xγi + γ

n∑

i=1

xγ−1i

λ + βγxγ−1i

+ (α − 1)n∑

i=1

xγi


= 0,

∂ ln L(α, λ, β, γ)∂γ

= −β

n∑

i=1

xγi · ln xi + β

n∑

i=1

xγ−1i (1 + γ ln xi)λ + βγxγ−1

i

+ (α − 1)βn∑

i=1

xγi · ln xi


= 0.

We can obtain the MLEs α̂, λ̂, β̂ and γ̂ numerically by solving the above four nonlinearequations via iterative techniques such as the Newton-Raphson method. Since the MLEs of un-known parameters are not given in closed forms, it is difficult to drive the exact distributions of


the MLEs. However, we can compute the approximate confidence intervals of parameters basedon the asymptotic distributions of their estimators via large sample approximation, where theMLEs can be treated as an approximately four variate normal distribution. Of course, the trueparameter vector is often supposed to be an interior point of the parameter space. Bootstrapresampling method is also a good method to obtain confidence intervals of all parameters.

To assess the finite sample properties of the NGLED, we conduct a simulation study tocompute the MLEs of the NGLED. We just consider parameter estimation under differentsample sizes n = 20, 40, 60 for the selected NGLED models. Table 1 represents average biasesand average mean square errors (MSEs) of 500 replications for different cases. From Table 1,overall, we observe that all biases and MSEs decrease as the sample size increases.

Table 1. The Estimated Biases and AMSEs of the MLEs

(α, λ, β, γ) n bias(α̂) bias(λ̂) bias(β̂) bias(γ̂)

MSE(α̂) MSE(λ̂) MSE(β̂) MSE(γ̂)

20 0.1258 −0.0457 0.0448 0.1340

2.0164 0.4525 0.1402 3.4402

(1.2,0.5,0.3,1.5) 40 0.0528 −0.0536 0.0642 0.0157

1.6475 0.2088 0.1397 2.4088

60 0.0490 −0.0478 0.0670 −0.0415

1.6080 0.2114 0.1387 2.1607

20 0.0388 0.9754 −0.2349 −0.0595

0.1258 18.590 0.6810 0.5765

(0.3,2,1,0.8) 40 0.0235 0.5029 −0.1834 −0.0539

0.1097 8.5594 0.7020 0.5764

60 0.0172 0.2619 −0.1645 −0.0483

0.1042 6.4111 0.7206 0.5824

4 Real Data Analysis

4.1 Example 1

In this subsection, a real data set is considered to illustrate how the NGLED model works inpractice. The data set is obtained from Linhart and Zuchini[7], and it is given in Table 2. Thisdata set represents the failure times of the air conditioning system of an air plane (in hours).The data set has been analyzed by many papers such as Gupta and Kundu[2], Mokhtari et al.[9]

and so on.

Table 2. Failure times of the Air Conditioning System

1 3 5 7 11 11 11 12 14 14

14 16 16 20 21 23 42 47 52 62

71 71 87 90 95 120 120 225 246 261

In order to illustrate that the NGLED can be a good lifetime model, we compare it withthe following nine different distributions:

(1) Exponential distribution (ED) with the pdf

f(x; λ) = λe−λx, x > 0.

(2) Reyleigh distribution (RD) with the pdf

f(x; β) = 2βxe−βx2, x > 0.


(3) Linear exponential distribution (LED) with the pdf

f(x; λ, β) = (λ + 2βx)e−(λx+βx2), x > 0.

(4) Weibull distribution (WD) with the pdf

f(x; β, γ) = βγxγ−1e−βxγ

, x > 0.

(5) Generalized exponential distribution (GED) with the pdf

f(x; α, λ) = αλe−λx(1 − e−λx)α−1, x > 0.

(6) Generalized Reyleigh distribution (GRD) with the pdf

f(x; α, β) = 2αβxe−βx2(1 − e−βx2

)α−1, x > 0.

(7) Exponential Weibull distribution (EWD) with the pdf

f(x; α, β, γ) = αβγxγ−1e−βxγ

(1 − e−βxγ

)α−1, x > 0.

(8) Generalized linear failure rate distribution (GLFRD) with the pdf

f(x; α, λ, β) = α(λ + 2βx)e−(λx+βx2)(1 − e−(λx+βx2))α−1, x > 0.

(9) Modified Weibull distribution (MWD) with the pdf

f(x; λ, β, γ) = (λ + βγxγ−1)e−(λx+βxγ), x > 0.

Table 3. The Associated K-S Distance Values and p-values for Different Models at Significance Level α = 0.05.

The model MLEs K-S p-value H

RD(β) β̂ = 7.3104 × 10−5 0.4954 3 × 10−7 0

LED(λ, β) λ̂ = 0.0164, β̂ = 3.4644 × 10−6 0.2184 0.0983 1

ED(λ) λ̂ = 0.0168 0.2132 0.1125 1

GRD(α, β) α̂ = 0.2486, β̂ = 2.3674 × 10−5 0.1972 0.1692 1

WD(β, γ) β̂ = 0.0215, γ̂ = 0.9468 0.1917 0.1935 1

MWD(λ, β, γ) λ̂ = 6.7798 × 10−5, β̂ = 0.0214, γ̂ = 0.9466 0.1917 0.1936 1

EWD(α, β, γ) α̂ = 0.8384, β̂ = 0.0161, γ̂ = 0.9832 0.1730 0.2954 1

GLFRD(α, λ, β) α̂ = 0.8064, λ̂ = 0.0144, β̂ = 4.9116 × 10−7 0.1722 0.3003 1

GED(α, λ) α̂ = 0.8089, λ̂ = 0.0145 0.1719 0.3020 1

NGLED(α, λ, β, γ) α̂ = 0.8348, λ̂ = 0.0055, β̂ = 0.0107, γ̂ = 0.9718 0.1710 0.3082 1

We implement three ordinary goodness-of-fit tests including the Kolmogonov-Smirnov (K-S) test, Cramer-von Mises test and the maximized likelihood approach to find out the best fitdistribution. It is well-known that the MLEs can provide better results for small sample. Themaximum likelihood estimates of all different models, the corresponding Kolmogonov-Smirnov(K-S) distances and p-values are presented in Table 3. All related estimated log-likelihoodvalues are listed in Table 4. The values of Cramer-von Mises test statistic are listed in Table 5.In all the tables, H = 0 means rejecting the null fitted distribution at specific significance levelα, while H = 1 represents accepting it at the same significance level.


Table 4. The Estimated Log-likelihood Values for Different Models

The Model log-likelihood

RD(β) −182.917

LED(λ, β) −152.792

ED(λ) −152.630

GRD(α, β) −155.042

WD(β, γ) −152.226

MWD(λ, β, γ) −152.226

EWD(α, β, γ) −152.167

GLFRD(α, λ, β) −152.211

GED(α, λ) −152.201

NGLED(α, λ, β, γ) −152.166

Table 5. The Associated Cramer-von Mises Distance Values for Different Models at Significance Level α = 0.05

The Model Test Statistics H

RD(β) 2.8187 0

LED(λ, β) 0.2261 0

EDλ) 0.2128 1

GRD(α, β) 0.3056 0

WD(β, γ) 0.1597 1

MWD(λ, β, γ) 0.1605 1

EWD(α, β, γ) 0.1223 1

GLFRD(α, λ, β) 0.1224 1

GED(α, λ) 0.1219 1

NGLED(α, λ, β, γ) 0.1194 1

According to the K-S test values in Table 3, we conclude that all suggested distributionsexcept for the RD can be deemed as good lifetime models for fitting given lifetime data at thesignificance level 0.05. The NGLED possesses the largest p value and the smallest K-S distance.Table 4 indicates that the NGLED can provide the largest log-likelihood value in all suggestedmodels. From Table 5, we can see that the NGLED displays the best performance from theviewpoint of Cramer-von Mises test at significance level α = 0.05. Overall, the NGLED is thebest candidate distribution in all compared distributions for this data set. Figure 8 is providedto compare the empirical reliability function with the theoretical reliability functions of allsuggested distributions.

4.2 Example 2

In this subsection, we analyze another real data to illustrate the desirable performance of theNGLED model in practice. The data set is cited from Lawless[5] and listed in Table 6. Thisdata set consists of failure times for 36 appliances subjected to an automatic life test. Kunduand Joarder[3] considered parameter estimation for the exponential distribution under type-IIhybrid progressively censored samples. Here, to illustrate that the NGLED can be a reasonablelifetime model, we compare it with all suggested distributions in Example 1.

Similarly, we implement Kolmogonov-Smirnov (K-S) test, Cramer-von Mises test and themaximized likelihood approach to compare the maximum likelihood estimates and the cor-responding p values of all suggested distributions are reported in Table 7. The related log-


Figure 8. The Empirical and Fitted Survival Function for Different Suggested Distributions

likelihood values are listed in Table 8. The values of Cramer-von Mises test statistic are listedin Table 9.

Table 6. Failure Times for 36 Appliances Subjected to an Automatic Life Test

11 35 49 170 329 381 708 958 1062 1167 1594 1925

1990 2223 2327 2400 2451 2471 2551 2565 2568 2694 2702 2761

2831 3034 3059 3112 3214 3478 3504 4329 6367 6976 7846 13403

Table 7. The Associated K-S Distance Values and p-values for Different Models at Significance Level α = 0.05

The Model MLEs K-S p-value H

RD(β) β̂ = 6.36 × 10−8 0.3240 7.29 × 10−4 0

LED(λ, β) λ̂ = 2.96 × 10−4, β̂ = 1.27 × 10−8 0.1641 0.2575 1

ED(λ) λ̂ = 3.63 × 10−4 0.1970 0.1064 1

GRD(α, β) α̂ = 0.3649, β̂ = 3.32 × 10−8 0.1900 0.1300 1

WD(β, γ) β̂ = 2.14 × 10−4, γ̂ = 1.0573 0.1641 0.2570 1

MWD(λ, β, γ) λ̂ = 1.99 × 10−5, β̂ = 1.89 × 10−4, γ̂ = 1.0616 0.1722 0.2100 1

EWD(α, β, γ) α̂ = 0.1599, β̂ = 1.93 × 10−3, γ̂ = 0.1049 0.5685 5.20 × 10−11 0

GLFRD(α, λ, β) α̂ = 0.3655, λ̂ = 2.18 × 10−8, β̂ = 3.33 × 10−8 0.1902 0.1293 1

GED(α, λ) α̂ = 0.9537, λ̂ = 3.52 × 10−4 0.2028 0.0892 1

NGLED(α, λ, β, γ) α̂ = 0.9772, λ̂ = 3.42 × 10−4, β̂ = 8.52 × 10−5, γ̂ = 0.7910 0.2019 0.0917 1

Table 8. The Estimated log-likelihood Values for Different Models

The Model log-likelihood

RD(β) −339.966

LED(λ, β) −321.175

ED(λ) −321.185

GRD(α, β) −321.272

WD(β, γ) −321.228

MWD(λ, β, γ) −321.279

EWD(α, β, γ) −442.523

GLFRD(α, λ, β) −321.288

GED(α, λ) −321.168

NGLED(α, λ, β, γ) −321.166


According to the K-S test values in Table 7, we see that all suggested distributions exceptfor the RD and the EWD can provide good fits for given lifetime data at the significance level0.05. We can see that the NGLED seems to be not the best choice in this case because its pvalue is not the largest. However, Table 8 indicates that the NGLED can provide the largestlog-likelihood value in all suggested models, which means that the NGLED performs best. Infact, the maximum likelihood approach can provide a more credible result for small sample size.From Cramer-von Mises distances in Table 9, we can see that all suggested distributions exceptthe RD and the EWD can provide good fits for given lifetime data at the level of significance0.01. Here, the reason we select the significance level 0.01 is because that all the suggesteddistributions cannot pass the Cramer-von Mises test at the level of significance 0.05. Althoughthe NGLED does not display the best performance in this case, overall, the NGLED can stillprovide a modest object of reference in all compared distributions in this example. The resultof NGLED is very close to that of GED and therefore it can be seemed as the best alternativedistribution for the data set apart from GED. Since different test approaches may result indifferent conclusions. We should refer to different results in practical applications. Figure 9 isdisplayed to contrast the empirical survival function with the theoretical survival functions ofall suggested distributions.

Table 9. The Associated Cramer-von Mises Distance Values for Different Models at Significance Level α = 0.01

The Model Test Statistics H

RD(β) 1.2296 0

LED(λ, β) 0.2307 1

ED(λ) 0.2995 1

GRD(α, β) 0.2644 1

WD(β, γ) 0.2472 1

MWD(λ, β, γ) 0.2458 1

EWD(α, β, γ) 3.0790 0

GLFRD(α, λ, β) 0.2642 1

GED(α, λ) 0.3169 1

NGLED(α, λ, β, γ) 0.3141 1

Figure 9. The Empirical and Fitted Survival Function for Different Suggested Distributions


5 Conclusion

We propose a new four-parameter GLED which can be seen as a new generalization for thelinear exponential distribution as some special cases, including the GLFRD, the EWD or theMWD and so on. Statistical properties and parameter estimations of the NGLED are studied.Finally, two real data sets are analyzed for illustration purpose. The results of the comparisonsfor different distributions show that the proposed NGLED can provide a better candidate thanother mentioned distributions for the given data sets.

Appendix A

We have the k-th moment of the NGLED as follows

μk =∫ ∞

0

xk · f(x; α, λ, β, γ) dx

=∫ ∞

0

xkα(λ + βγxγ−1)e−(λx+βxγ)(1 − e−(λx+βxγ))α−1dx.

Using the following expansion of (1 − e−(λx+βxγ))α−1 given by

(1 − e−(λx+βxγ))α−1 =∞∑

n=0

(α−1n )(−1)ne−n(λx+βxγ),

then we have

μk =∫ ∞

0

xkα(λ + βγxγ−1)e−(λx+βxγ)∞∑

n=0

(α−1n )(−1)ne−n(λx+βxγ)dx

=∫ ∞

0

xkα(λ + βγxγ−1)∞∑

n=0

(α−1n )(−1)ne−(n+1)(λx+βxγ)dx

=∫ ∞

0


n=0

(α−1n )(−1)ne−(n+1)λxe−(n+1)βxγ

dx

=∫ ∞

0


n=0

(α−1n )(−1)ne−(n+1)λx

∞∑

s=0

(−(n + 1)β)s

s!xsγdx

=α

∫ ∞

0

∞∑

n=0

∞∑

s=0

(α−1n )(−1)n+s

· (n + 1)sβs

s!e−(n+1)λx(λxsγ+k + βγxsγ+k+γ−1)dx

=∞∑

n=0

∞∑

s=0

(−1)n+s(αn) · βs · (n + 1)(1−γ)s−k−γ

λsγ+ks!

·[(n + 1)γ−1 · Γ(sγ + k + 1) +

βγ · Γ(sγ + k + γ)λγ

].

Appendix B

We start the moment generating function of the NGLED as follows


M(t) =E(etX) =∫ ∞

0

etx · f(x; α, λ, β, γ)dx

=∫ ∞

0

etxα(λ + βγxγ−1)e−(λx+βxγ)(1 − e−(λx+βxγ))α−1dx.

Similarly, using the following expansion of (1 − e−(λx+βxγ))α−1 given by

(1 − e−(λx+βxγ))α−1 =∞∑

n=0

(α−1n )(−1)ne−n(λx+βxγ),

then, we have

M(t) =∫ ∞

0

etxα(λ + βγxγ−1)e−(λx+βxγ)∞∑

n=0

(α−1n )(−1)ne−n(λx+βxγ)dx

=∞∑

n=0

(αn)(−1)n

∫ ∞

0

(λ + βγxγ−1)e−((n+1)λ−t)xe−(n+1)βxγ

dx

=∞∑

n=0

(αn)(−1)n

∫ ∞

0

(λ + βγxγ−1)e−((n+1)λ−t)x ·∞∑

s=0

(−(n + 1)β)s

s!xsγdx

=∞∑

n=0

∞∑

s=0

(αn)(−1)n+s · (n + 1)sβs

s!

·∫ ∞

0

e−((n+1)λ−t)x(λxsγ + βγxsγ+γ−1)dx

=∞∑

n=0

∞∑

s=0

(αn)(−1)n+s (n + 1)sβs

s!

·[ λ · Γ(sγ + 1)((n + 1)λ − t)sγ+1

+βγ · Γ(sγ + γ)

((n + 1)λ − t)sγ+γ

], t ≤ λ.

For t > λ, the moment generating function does not exist in this case.

Acknowledgements. During this modification process, we find that our proposed modelprocess exactly analogous form with Zaindin and Sarhan[16]. However, we conducted our re-search independent of their research. The authors appreciate sincerely the reviewers’ valuablecomments and suggestions on our manuscript, which have helped us improve the presentationof the paper essentially.

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A new generalized linear exponential distribution and its applications