# Exponentiated Discrete Lindley Distribution: Properties and Applications

In this article, the exponentiated discrete Lindley distribution is presented and studied. Some important distributional properties are discussed. Using the maximum likelihood method, estimation of the model parameters is investigated. Furthermore, simulation study is performed to observe the performance of the estimates. Finally, the model with two real data sets is examined.

## Authors

• 3 publications
• 2 publications
• 1 publication
10/20/2021

08/22/2018

### Bivariate Discrete Inverse Weibull Distribution

In this paper, we propose a new class of bivariate distributions, called...
01/16/2018

### On a bimodal Birnbaum-Saunders distribution with applications to lifetime data

The Birnbaum-Saunders distribution is a flexible and useful model which ...
03/01/2019

### The wrapped xgamma distribution for modeling circular data appearing in geological context

The technique of wrapping of a univariate probability distribution is ve...
05/31/2021

### On some properties of the bimodal normal distribution and its bivariate version

In this work, we derive some novel properties of the bimodal normal dist...
11/02/2021

### Discrete Bilal distribution with right-censored data

This paper presents inferences for the discrete Bilal (DB) distribution ...
05/16/2020

### Transforming variables to central normality

Many real data sets contain features (variables) whose distribution is f...
##### This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

## 1 Introduction

Statistical (lifetime) distributions are commonly applied to describe and predict real world phenomena. Several classical distributions have been extensively used over the past decades for modeling data in several fields such as engineering, medicine, finance, biological and actuarial science. Lindley distribution (LiD) is one of the most important lifetime distributions, it has some nice properties to be used in lifetime data analysis, especially in applications modeling stress-strength model (see, Lindley (1958)). This distribution can be shown as a mixture of exponential and gamma distributions. The random variable (RV)

is said to have LiD with one scale parameter

, if the cumulative distribution function (CDF) and the probability density function (PDF) are given by

 ∏(z;a)=1−e−az(1+aza+1);  z>0, (1)

and

 π(z;a)=a21+a(z+1)e−az;  z>0, (2)

respectively. Due to its wide applicability in many areas, several works aimed at extending the LiD become very important. See, Ghitany et al. (2008a, 2008b, 2011, 2013), Zakerzadeh and Dolati (2009), Mahmoudi and Zakerzadeh (2010), Jodrá (2010), Nadarajah et al. (2011), Bakouch et al. (2012), Merovci (2013), Shanker and Mishra (2013), Shanker et al. (2013), Merovci and Elbatal (2014), Merovci and Sharma (2014), Liyanage and Pararai (2014), Pararai et al. (2015), Sharma et al. (2015), Nedjar and Zeghdoudi (2016), Zeghdoudi and Nedjar (2016, 2017), Özel et al. (2016), Elbatal et al. (2016), Altun et al. (2017), Mahmoud (2018), Jehhan et al (2018), among others.

Furthermore, some discrete versions of the LiD have been presented in the statistical literature because in several cases, lifetimes need to be recorded on a discrete scale rather than on a continuous analogue. So, discretizing continuous distributions has received much attention in the statistical literature. See, Sankaran (1970), Ghitany and Al-Mutairi (2009), Calderín-Ojeda and Gómez-Déniz (2013), Bakouch et al. (2014), Tanka and Srivastava (2014), Munindra et al. (2015), Kus et al. (2018), among others.

Also, several discrete distributions have been presented in the literature. See, Roy (2003, 2004), Inusah and Kozubowski (2006), Krishna and Pundir (2009), Gómez-Déniz (2010), Bebbington et al. (2012), Nekoukhou et al. (2013), Vahid and Hamid (2015), Alamatsaz et al. (2016), Chandrakant et al. (2017), among others.

Although there are a number of discrete distributions in the literature, there is still a lot of space left to develop new discretized distribution that is suitable under different conditions. So, in this article, we introduce a flexible discrete distribution called, the exponentiated discrete Lindley distribution (EDLiD), because the discrete Lindley distribution (DLiD) does not supply enough flexibility for analysis different types of lifetime data.

The article is organized as follows. In Section 2, we introduce the EDLiD. Different statistical properties are studied in Section 3 . The estimation of the model parameters by maximum likelihood is performed in Section 4. In Section 5, simulation study is presented. Moreover, two applications to real data illustrate the potentiality of the EDLiD. Finally, Section 6 provides some conclusions.

## 2 The EDLiD

Gómez-Déniz and Calderín-Ojeda (2011) introduced the DLiD. The RV is said to have DLiD with a parameter if the CDF and the probability mass function (PMF) are given by

 W(y;a)=1−ay+1+[(2+y)ay+1−1]loga1−loga;  y∈N0={0,1,2,3,...}, (3)

and

 w(y,a)=ay1−loga[aloga+(1−a)(1−logay+1)];  y∈N0, (4)

respectively. In the context of lifetime distributions with CDF , the most widely used generalization technique is the exponentiated-W. Using this method, for , the CDF of the exponentiated-W class is given by

 F(y;a,b)=[W(y;a)]b, (5)

(see, Lehmann (1952)). Therefore, the RV  is said to have EDLiD with shape parameter and scale parameter if the CDF and the reliability function are given by

 F(x;a,b)=Λ(x+1;a,b)(1−loga)b;\ \ x∈N0, (6)

and

 R(x;a,b)=(1−loga)b−Λ(x+1;a,b)(1−loga)b;\ \ x∈N0, (7)

respectively, where

 Λ(x;a,b)=(1−ax+[(1+x)ax−1]loga)b. (8)

Further, the PMF of the EDLiD is given by

 f(x;a,b)=1(1−loga)b[Λ(x+1;a,b)−Λ(x;a,b)] ;  x∈N0, (9)

where . Figure 1 shows the plots of the PMF for various values of the model parameters.

 Figure 1. The PMF of the EDLiD for various values of the parameters.

From Figure 1, we note that the EDLiD can be take different shapes depending on the values of the parameters. Moreover, the hazard rate function (Hrf) can be expressed as

 h(x;a,b)=Λ(x+1;a,b)−Λ(x;a,b)(1−loga)b−Λ(x+1;a,b);\ \ x∈N0. (10)

Figure 2 shows the plots of the Hrf for various values of the model parameters.

 Figure 2. The Hrf of the EDLiD for various values of the parameters.

From Figure 2, it is clear that the Hrf can be increasing, decreasing, bathtub and upside-down bathtub shaped. So, the EDLiD can be suitable for modeling various data sets. Also, the reversed hazard rate function (Rhrf) of the EDLiD can be expressed as follows

 r(x;a,b)=1−Λ(x;a,b)Λ(x+1;a,b);\ \ x∈N0. (11)

Figure 3 shows the plots of the Rhrf for various values of the model parameters.

 Figure 3. The Rhrf of the EDLiD for various values of the parameters.

## 3 Different Properties

### 3.1 Moments

Assume non-negative RV EDLiD. Then, the th moment, say , is given by

 ϖ′r =∞∑x=0xrf(x;a,b)=∞∑x=1[xr−(x−1)r]R(x;a,b) =1(1−loga)b∞∑x=1[xr−(x−1)r][(1−loga)b−Λ(x+1;a,b)]. (12)

Using Equation (12), we can get the mean (

) and the variance (

) of the random variable as follows

 ζ=1(1−loga)b∞∑x=1[(1−loga)b−Λ(x+1;a,b)], (13)

and

 Υ=1(1−loga)b∞∑x=1[2x−1][(1−loga)b−Λ(x+1;a,b)]−ζ2, (14)

respectively. Since th moment is not in a closed form, then and can only be numerically evaluated. Tables 1 and 2 obtain and of the EDLiD for different values of the model parameters respectively.

 Table 1.\ The ζ of the EDLiD for different % values of the parameters a and b.
 Table 2.\ The\ Υ of the EDLiD for different % values of the parameters a and b.

Depending on the model parameters, Tables 1 and 2 obtain that and are increasing when is constant (increasing) and

is increasing (constant). Furthermore, the skewness (

) and the kurtosis (

) can be calculated as follows and . Tables 3 and 4 obtain the and of the EDLiD for different values of the model parameters respectively.

 Table 3.\ The Ξ of the EDLiD for different values% of the parameters a and b.
 Table 4.\ The Θ of the EDLiD for different % values of the parameters a and b.

Tables 3 and 4 obtain that and are decreasing when is constant and is increasing. On the other hand, we can get the probability generating function (PGF) of the RV as a form

 ΩX(t) =∞∑x=0txf(x;a,b) =R(0)+(t−1)R(1)+(t2−t)R(2)+(t3−t2)R(3)+... =1+(t−1)(1−loga)b∞∑x=1tx−1[(1−loga)b−Λ(x+1;a,b)]. (15)

Using Equation (15), we can get and of the RV as a form and .

In order to study the ageing behavior of a component or a system of components there have been defined several measures in the reliability and survival analysis literature. The is a helpful tool to model and analyze the burn-in and maintenance policies. In the discrete setting, is defined as

 ς(i)=E(T−i|T≥i)=1R(i)l∑j=i+1R(j) ;  i∈N0, (16)

where . If the RV EDLiD, then the can be expressed as follows

 ς(i)=1(1−loga)b−Λ(i+1;a,b)l∑j=i+1[(1−loga)b−Λ(j+1;a,b)]. (17)

Another measure of interest in survival analysis is . It measures the time elapsed since the failure of given that the system has failed sometime before . In the discrete setting, is defined as

 ς∗(i)=E(i−T|T

where (see, Goliforushani and Asadi (2008)). If the RV EDLiD, then the can be represented as follows

 ς∗(i)=1Λ(i;a,b)i∑m=1Λ(m;a,b). (19)

For we get .

Lemma 1. The mean of the RV EDLiD can be expressed as

 ζ=i−Λ(i;a,b)(1−loga)bς∗(i)+(1−loga)b−Λ(i+1;a,b)(1−loga)bς(i) ;  i∈N0.

Proof. It is easy to prove this Lemma by using the following Equation

 ζ=ς(0)=l∑j=1R(j;a,b)=i∑j=1R(j;a,b)+l∑j=i+1R(j;a,b).

Lemma 2. The Rhrf and the are related as follows

 r(i;a,b)=1−ς∗(i+1;a,b)+ς∗(i;a,b)ς∗(i;a,b) ;  i∈N0−{0}. (20)

Proof.

 F(i;a,b)ς∗(i+1;a,b)−F(i−1;a,b)ς∗(i;a,b) =i+1∑j=1F(j−1;a,b)−i∑j=1F(j−1;a,b) =F(i;a,b).

Dividing both sides of this Equation by , and noting that , we get the required result.

### 3.3 Stress-strength (S-S∗) analysis

S-S analysis has been used in mechanical component design. The probability of failure is based on the probability of S exceeding S. Assume that both S and S are in the positive domain. The expected reliability () can be calculated by

 R∗=P[XS≤XS∗]=∞∑x=0fXS(x)RXS∗(x) \ . (21)

If EDLiD and EDLiD, then

 R∗=∑∞x=0[Λ(x+1;a1,b1)−Λ(x;a1,b1)][(1−loga2)b2−Λ(x+1;a2,b2)](1−loga1)b1(1−loga2)b2. (22)

From Equation (22), it is clear that the value of does not depend only on the values of the model parameters.

### 3.4 Order statistics (Os) and L-moment (Lm) statistics

Let , be a random sample from the EDLiD, and let be their corresponding Os. Then, the CDF of th Os for an integer value of can be expressed as

 Fi:n(x;a,b) =n∑k=i(nk)[Fi(x;a,b)]k[1−Fi(x;a,b)]n−k =n∑k=in−k∑j=0⊝(n,k)(j)Λ(x+1;a,b(k+j))(1−loga)b(k+j), (23)

where . Furthermore, the PMF of the th Os can be expressed as

 fi:n(x;a,b)=n∑k=in−k∑j=0⊝(n,k)(j)[Λ(x+1;a,b(k+j))−Λ(x;a,b(k+j))](1−loga)b(k+j). (24)

So, the moments of can be written as

 E(Xvi:n)=∞∑x=0n∑k=in−k∑j=0⊝(n,k)(j)xv[Λ(x+1;a,b(k+j))−Λ(x;a,b(k+j))](1−loga)b(k+j). (25)

On the other hand, Hosking (1990) has defined the L-moments (Lms) to summaries theoretical distribution and observed samples. He has shown that the Lms have good properties as measure of distributional shape and are useful for fitting distribution to data. Lms are expectation of certain linear combinations of Os. The Lms of    can be expressed as follows

 Δs=1ss−1∑j=0(−1)j(s−1j)E(Xs−j:s). (26)

Since Hosking has defined the Lms of to be the quantities. Then, we can propose some statistical measures such as Lm of mean , Lm coefficient of variation , Lm coefficient of skewness , and Lm coefficient of kurtosis .

## 4 Estimation

In this section, we determine the maximum likelihood estimates (MLEs) of the model parameters from complete samples. Assume be a random sample of size from the EDLiD(). The log-likelihood function () can be expressed as

 L(x;a,b)=−nblog(1−loga)+n∑i=1log[Λ(x+1;a,b)−Λ(x;a,b)]. (27)

By differentiating Equation (27) with respect to the parameters and , we get the normal nonlinear likelihood equations as follows

 nˆbˆa(1−logˆa)+ˆbn∑i=1[V1(xi+1;ˆa)]ˆb−1V2(xi+1;ˆa)−[V1(xi;ˆa)]ˆb−1V2(xi;ˆa)Λ(xi+1;ˆa,ˆb)−Λ(xi;ˆa,ˆb)=0, (28)

and

 −nlog(1−logˆa)+n∑i=1Λ(xi+1;ˆa,ˆb)log(V1(xi+1;ˆa))−Λ(xi;ˆa,ˆb)log(V1(xi;ˆa))Λ(xi+1;ˆa,ˆb)−Λ(xi;ˆa,ˆb)=0, (29)

respectively, where , and Analytical software is required to get the values of the model parameters.

## 5 Applications

### 5.1 Simulation results

In this section, we obtain the behavior of the MLEs of the EDLiD for a sample size using a simulation study. At first, to generate a RV from the EDLiD, we generate the value from the continuous ELiD. Then, discretize this value to obtain . The steps for a simulation study: choose the initial values of the model parameters, say EDLiD(), generate samples of size 25, 50, 100, 150, 200, 250, 300, 350, 400, 450, 500, 550, 600, 650, 700, 750, compute the MLE’s for the samples, say for . Finally, compute the average of biases and the average of mean squared errors (MSE(.)). Figure 4 shows how the biases and MSE vary with respect to .

 Figure 4. The plots of bias(a), bias(b), MSE(a) and MSE(b) versus n = 25, 50, 100,150, 200, …,750.

From Figure 4, it is clear that the biases and the MSEs of the estimated parameters while growing. So, the MLE is a good method for estimating the model parameters.

### 5.2 Data analysis

In this section, we illustrate the importance of the EDLiD using two real data sets.

The first data set (I): represents the number of women who are working on shells for 5 weeks discussed in Consul and Jain (1973). We shall compare the fits of the EDLiD with some competitive models such as discrete generalized exponential second type (DGE), discrete Weibull (DW), discrete Lindley (DLi), discrete Pareto (DPa) and Poisson (P) distributions.

The second data set (II): represents the counts of cysts of kidneys using steroids. This data set originated from a study Chan et al. (2009). We shall compare the fits of the EDLiD with some competitive models such as DW, discrete Burr-XII (DB-XII), discrete Lomax (DLo), geometric (Geo), DLi, P and discrete Rayleigh (DR) distributions.

The fitted models are compared using some criteria namely, the maximized log-likelihood (), Akaike Information Criterion (), Correct Akaike Information Criterion (), Bayesian Information Criterion (), Hannan-Quinn Information Criterion (), chi-square () and its P-value.

For the data set (I),

Tables 5 and 6 obtain the MLEs with their corresponding standard errors (Se(.)), as well as

, AIC, CAIC, BIC, HQIC,

, degree of freedom (d.f), observed frequency (OF), expected frequency (EF) and P-values respectively.

 Table 5. The MLEs with their corresponding Se for data set I.
 Table 6. The goodness of fit tests for data set I.

From Table 6, it is clear that the EDLiD is the best distribution among all tested distributions, because it has the smallest value among , AIC, CAIC, BIC, HQIC and , as well as it has the largest P-value. Figure 5 shows the fitted PMFs for data set I, which support the results in Table 6.

 Figure 5. The fitted PMFs for data set I.

For the data set (II), Tables 7 and 8 obtain the MLEs with their corresponding Se, as well as , AIC, CAIC, BIC, HQIC, OF, EF, d.f, and P-values respectively.

 Table 7. The MLEs with their corresponding Se for data set II.
 Table 8. The goodness of fit tests for data set II.

From Table 8, it is clear that the EDLiD is the best distribution among all tested models. Figure 6 shows the fitted PMFs for data set II, which support the results in Table 8.

 Figure 6. The fitted PMFs for data set II.

## 6 Conclusions

A two-parameter EDLiD has been proposed. Its various distributional properties have been discussed. It was found that the proposed distribution has a simple structure, is more flexible and has a longer tail than the DLiD and other discrete distributions in modeling data from different fields. In the future, we will discuss the bivariate and multivariate extensions of this distribution.

## References

• [1] Alamatsaz, M., Dey, H., Dey, S., Harandi, T., and Shams, S., (2016). Discrete generalized Rayleigh distribution. Pakistan journal of statistics, 32(1), 1-20.
• [2]

Altun, G., Alizadeh, M., Altun, E., and Özel, G., (2017). Odd Burr Lindley distribution with properties and applications. Hacettepe journal of statistics and mathematics, 46 (2), 255-276.

• [3] Bakouch, H. S., Aghababaei, M., and Nadarajah, S., (2014). A new discrete distribution. Statistics, 48(1), 200-240.
• [4] Bakouch, H. S., Al-Zahrani, B. M., Al-Shomrani, A. A., Marchi, V. A., and Louzada, F., (2012). An extended Lindley distribution. Journal of the Korean statistical society, 41, 75-85.
• [5] Bebbington, M., Lai, C. D., Wellington, M., and Zitikis, R., (2012). The discrete additive Weibull distribution: a bathtub-shaped hazard for discontinuous failure data. Reliability engineering and system safety, 106, 37-44.
• [6] Calderín-Ojeda, E., and Gómez-Déniz, E., (2013). An extension of the discrete Lindley distribution with applications. Journal of the Korean statistical society, 42, 371-373.
• [7] Chan, S., Riley, P. R., Price, K. L., McElduff, F., and Winyard, P. J., (2009). Corticosteroid-induced kidney dysmorphogenesis is associated with deregulated expression of known cystogenic molecules, as well as indian hedgehog. American journal of physiology-renal physiology, 298(2), 346-356.
• [8] Chandrakant, K., Yogesh, M. T., and Manoj, K. R., (2017). On a discrete analogue of linear failure rate distribution. American journal of mathematical and management sciences, 36(3), 229-246.
• [9]

Consul, P.C., and Jain, G. G., (1973). A generalization of the Poisson distribution. Technometrics, 15(4), 791-799.

• [10] Elbatal, I., Diab, L. S., and Elgarhy, M., (2016). Exponentiated quasi Lindley distribution. International journal of reliability and applications, 17(1), 1-19.
• [11] Ghitany, M. E., Al-Mutairi, D. K., and Nadarajah, S., (2008a). Zero-truncated Poisson Lindley distribution and its application. Mathematics and computers in simulation, 79(3), 279-287.
• [12] Ghitany, M. E., Al-Mutairi, D. K., Balakrishhnan, N., and Al-Enezi, L. J., (2013). Power Lindley distribution and associated inference. Computational statistics and data analysis, 64, 20-33.
• [13] Ghitany, M. E., Alqallaf, F., Al-Mutairi, D. K., and Husain, H. A., (2011). A two-parameter weighted Lindley distribution and its applications to survival data. Mathematics and computers in simulation, 81(6), 1190-1201.
• [14] Ghitany, M. E., and Al-Mutairi, D. K., (2009). Estimation methods for the discrete Poisson Lindley distribution. Journal of statistical computation and simulation, 79(1), 1-9.
• [15] Ghitany, M. E., Atieh, B., and Nadarajah, S., (2008b). Lindley distribution and its application. Mathematics and computers in simulation, 78(4), 493-506.
• [16] Goliforushani, S., and Asadi, M., (2008). On the discrete mean past lifetime. Metrika, 68, 209-217.
• [17]

Gómez-Déniz, (2010). Another generalization of the geometric distribution. Test, 19(2), 399-415.

• [18] Gómez-Déniz, E., and Calderín-Ojeda, E., (2011). The discrete Lindley distribution: properties and applications. Journal of statistical computation and simulation, 81(11), 1405–1416.
• [19] Hosking, J. R., (1990). L-moments: analysis and estimation of distributions using linear combinations of order statistics. Journal of the Royal statistical society, 52(B), 105-124.
• [20] Inusah, S., and Kozubowski, T. J., (2006). A discrete analogue of the Laplce distribution. Journal of statistical planning and infernce, 136, 1090-1102.
• [21] Jehhan, A., Mohamed, I. , Eliwa, M. S., Al-mualim, S., and Yousof, H. M., (2018). The two-parameter odd Lindley Weibull lifetime model with properties and applications. International journal of statistics and probability, 7, (4), 57-68.
• [22] Jodrá, P., (2010). Computer generation of random variables with Lindley or Poisson-Lindley distribution via the Lambert W function. Mathematics and computers in simulation, 81(4), 851-859.
• [23] Krishna, H., and Pundir, P. S., (2009). Discrete Burr and discrete Pareto distributions. Statistical methodology, 6, 177-188.
• [24] Kus, C., Akdogan,Y., Asgharzadeh, A., Kinaci, I., and Karakaya, K., (2018). Binomial-discrete Lindley distribution. Communications faculty of sciences university of Ankara series A1-mathematics and statistics, 68(1), 401-411.
• [25] Lehmann, E. L., (1952). The power of rank tests. Annals of mathematical statistics, 24, 23-43.
• [26]

Lindley, D. V., (1958). Fiducial distributions and Bayes theorem. Journal of the Royal statistical society, 20(B), 102-107.

• [27] Liyanage, W., and Pararai, M., (2014). A generalized power Lindley distribution with applications. Asian journal of mathematics and applications, 18, 1-23.
• [28] Mahmoud, E., (2018). Logarithmic inverse Lindley distribution: model, properties and applications. Journal of King Saud university science. To appear.
• [29] Mahmoudi, E., and Zakerzadeh, H., (2010). Generalized Poisson-Lindley distribution. Communications in statistics: theory and methods, 39(10), 1785-1798.
• [30] Merovci, F., (2013). Transmuted Lindley distribution. International journal of open problems in computer science and mathematics, 6(2), 63-72.
• [31] Merovci, F., and Elbatal, I., (2014). Transmuted Lindley-geometric distribution and its applications. Journal of statistics applications and probability, 3(1), 77-91.
• [32] Merovci, F., and Sharma, V. K., (2014). The Beta-Lindley distribution: properties and applications. Journal of applied mathematics, http://dx.doi.org/10.1155/2014/198951.
• [33] Munindra, B., Krishna, R. S., and Junali, H., (2015). A study on two parameter discrete quasi Lindley distribution and its derived distributions. International journal of mathematical archive, 6(12), 149-156.
• [34] Nadarajah, S., Bakouch, H. S., and Tahmasbi, R., (2011). A generalized Lindley distribution. Sankhya B, 73, 331-359.
• [35]

Nekoukhou, V., Alamatsaz, M. H., and Bidram, H., (2013). Discrete generalized exponential distribution of a second type. Statistics, 47 (4), 876-887.

• [36] Nedjar, S., and Zeghdoudi, H., (2016). On gamma Lindley distribution: proprieties and simulations. Journal of computational and applied mathematics, 298, 167-174.
• [37] Özel, G., Alizadeh, M., Cakmakyapan, S., Hamedani, G., Ortega, E. M., and Cancho, G., (2017). The odd log-logistic Lindley Poisson model for lifetime data. Communications in statistics: simulation and computation, 46(8), 6513-6537.
• [38] Pararai, M., Warahena-Liyanage, G., and Oluyede, B. O., (2015). A new class of generalized power Lindley distribution with applications to lifetime data. Theoretical mathematics and applications, 5, 53-96.
• [39]

Roy, D., (2003). The discrete normal distribution. Communications in statistics: Theory and methods, 32, 1871-1883.

• [40] Roy, D., (2004). Discrete Rayleigh distribution. IEEE transactions on reliability, 53(2), 255-260.
• [41] Sankaran, M., (1970). The discrete Poisson-Lindley distribution. Biometrics, 26(1), 145-149.
• [42] Shanker, R., and Mishra, A., (2013). A quasi Lindley distribution. African journal of mathematics, 6(4), 64-71.
• [43] Shanker, R., Sharma, S., and Shanker, R., (2013). A two-parameter Lindley distribution for modeling waiting and survival times data. Applied mathematics, 4, 363-368.
• [44] Sharma, V., Singh, S., Singh, U., and Agiwal, V., (2015). The inverse Lindley distribution: a stress-strength reliability model with applications to head and neck cancer data. Journal of industrial and production engineering, 32(3), 162-173.
• [45] Tanka, R. A., and Srivastava, R. S., (2014). Size-biased discrete two parameter Poisson-Lindley distribution for modeling and waiting survival times data. Journal of mathematics, 10(1), 39-45.
• [46] Vahid, N., and Hamid, B., (2015). The exponentiated discrete Weibull distribution. Sort, 39 (1), 127-146.
• [47] Zakerzadeh, H., and Dolati, A., (2009). Generalized lindley distribution. Journal of mathematical extension, 3(2), 13-25.
• [48] Zeghdoudi, H., and Nedjar, S., (2016). Gamma Lindley distribution and its application. Journal of applied probability and statistics, 11(1), 129-138.
• [49] Zeghdoudi, H., and Nedjar, S., (2017). On Poisson pseudo Lindley distribution: Properties and applications. Journal of probability and statistical science, 15(1), 19-28.