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 stressstrength 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
(1) 
and
(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 AlMutairi (2009), CalderínOjeda and GómezDéniz (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), GómezDéniz (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
GómezDéniz and CalderínOjeda (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
(3) 
and
(4) 
respectively. In the context of lifetime distributions with CDF , the most widely used generalization technique is the exponentiatedW. Using this method, for , the CDF of the exponentiatedW class is given by
(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
(6) 
and
(7) 
respectively, where
(8) 
Further, the PMF of the EDLiD is given by
(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
(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 upsidedown 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
(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 nonnegative RV EDLiD. Then, the th moment, say , is given by
(12) 
Using Equation (12), we can get the mean (
) and the variance (
) of the random variable as follows(13) 
and
(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.
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.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
(15) 
Using Equation (15), we can get and of the RV as a form and .
3.2 Mean residual lifetime () and mean past lifetime ()
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 burnin and maintenance policies. In the discrete setting, is defined as
(16) 
where . If the RV EDLiD, then the can be expressed as follows
(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
(18) 
where (see, Goliforushani and Asadi (2008)). If the RV EDLiD, then the can be represented as follows
(19) 
For we get .
Lemma 1. The mean of the RV EDLiD can be expressed as
Proof. It is easy to prove this Lemma by using the following Equation
Lemma 2. The Rhrf and the are related as follows
(20) 
Proof.
Dividing both sides of this Equation by , and noting that , we get the required result.
3.3 Stressstrength (SS) analysis
SS 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
(21) 
If EDLiD and EDLiD, then
(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 Lmoment (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
(23) 
where . Furthermore, the PMF of the th Os can be expressed as
(24) 
So, the moments of can be written as
(25) 
On the other hand, Hosking (1990) has defined the Lmoments (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
(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 loglikelihood function () can be expressed as
(27) 
By differentiating Equation (27) with respect to the parameters and , we get the normal nonlinear likelihood equations as follows
(28) 
and
(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 = 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 BurrXII (DBXII), discrete Lomax (DLo), geometric (Geo), DLi, P and discrete Rayleigh (DR) distributions.
The fitted models are compared using some criteria namely, the maximized loglikelihood (), Akaike Information Criterion (), Correct Akaike Information Criterion (), Bayesian Information Criterion (), HannanQuinn Information Criterion (), chisquare () and its Pvalue.
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 Pvalues 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 Pvalue. 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 Pvalues 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 twoparameter 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.
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