Generalized Marshall Olkin Inverse Lindley Distribution with Applications

In this article, a new generalization of the inverse Lindley distribution is introduced based on Marshall-Olkin family of distributions. We call the new distribution, the generalized Marshall-Olkin inverse Lindley distribution which offers more flexibility for modeling lifetime data. The new distribution includes the inverse Lindley and the Marshall-Olkin inverse Lindley as special distributions. Essential properties of the generalized Marshall-Olkin inverse Lindley distribution are discussed and investigated including, quantile function, ordinary moments, incomplete moments, moments of residual and stochastic ordering. Maximum likelihood method of estimation is considered under complete, Type-I censoring and Type-II censoring. Maximum likelihood estimators as well as approximate confidence intervals of the population parameters are discussed. A comprehensive simulation study is done to assess the performance of estimates based on their biases and mean square errors. The notability of the generalized Marshall-Olkin inverse Lindley model is clarified by means of two real data sets. The results showed the fact that the generalized Marshall-Olkin inverse Lindley model can produce better fits than power Lindley, extended Lindley, alpha power transmuted Lindley, alpha power extended exponential and Lindley distributions.


Introduction
In the last decade, the general method of adding a shape parameter to expand a family of distributions was introduced by Marshall et al. [Marshall and Olkin (1997)]. This family is called the Marshall-Olkin (MO)-G class. The cumulative distribution function (cdf) and the probability density function (pdf) of the MO-G class are defined as follows ( ) ( ) and, where, 0, is the survival function and ( ) W x is the baseline distribution. The parameterα is known as a tilt parameter and interpreted α in terms of the behavior of the hazard rate function (hrf) of X. This ratio is increasing in X for 1 α ≥ and decreasing in X for (0,1) α ∈ (see Nanda et al. [Nanda and Das (2012)]).
The generalization of the MO-G family proposed by Jayakumar et al. [Jayakumar and Mathew (2008)] through Lehmann alternative 1 approach by exponentiating the MO survival function (sf) as where 0, b > is an additional shape parameter. When The pdf corresponding to Eq. (3) is given by For more information about Marshall-Olkin family see Barakat et al. [Barakat, Ghitany and AL-Hussaini (2009) The Lindley distribution and its applications have been discussed by Ghitany et al. [Ghitany, Atieh and Nadadrajah (2008)] and showed that the Lindley distribution is a better fit than the exponential distribution based on the waiting time at the bank for service. Sharma et al. [Sharma, Singh, Singh et al. (2015)] proposed the inverse Lindley (IL) distribution by using the transformation 1 , where Y has the pdf in Eq. (5) and cdf in Eq. (6), with the following pdf 2 3 1 ( ; ) , , 0.
For 1 b = , the GMOIL distribution reduces to MOIL distribution. For 1 b α = = , the GMOIL distribution reduces to IL (see Ghitany et al. [Ghitany, Atieh and Nadadrajah (2008)]). The sf and hrf of the GMOIL distribution are respectively, given by Plots of the pdf and hrf of the GMOIL distribution are displayed in Fig. 1, for different values of parameters. As seen from Fig. 1, the shapes of the pdf take different forms. Also, it is clear that the shapes of the hrf are decreasing, increasing and up-side down shaped at some selected values of parameters.

Main properties
In this section, we obtain some important statistical properties of the GMOIL distribution such as quantile function, ordinary and incomplete moments, moment generating function, moments of the residual and reversed residual lives and stochastic ordering.

Quantile function
Quantiles are essential for estimation and simulation. The quantile function, say , is obtained by inverting Eq. (10) as follows Multiply both sides by (1 ) (1 )e θ θ − + + , then we have the Lambert equation Hence, we have the negative Lambert W function of the real argument i.e., where (0,1) u ∈ , and 1 (.) W − is the negative Lambert W function.

Moments and incomplete moments
The th s moment about zero for the GMOIL distribution is derived. To obtain the th s moment, firstly explicit expression for the pdf in Eq. (9) is obtained. Since, the binomial expansion, for real non-integer value of m, is given by Then by employ Eq. (18) in Eq. (9), then 1 2 Apply the generalized binomial expansion, in Eq.
Again, using the binomial expansion in Eq. (20) , , 3 , 0 0 1 ( ; ) , where, 2 , , Hence, the th s moment of the GMOIL distribution is obtained as follows which leads to , , 2 1 , 0 0 The th s central moment ( s µ ) of X is given by Recall the Taylor is the upper incomplete gamma function. Bonferroni and Lorenz curves measures of in-equality are widely used in various fields such as survival analysis, demography and insurance. These measures are the main applications of the first incomplete moment.

Residual and reversed residual life functions
Here, the th k moment of the residual lifetime (MRL) of a random variable X is defined as follows Employing the binomial expansion and Eq. (21) in Eq. (28), then the th k MRL of the GMOIL distribution is derived as follows ( 1) , , 3 0 , 0 0 After simplification, the th k MRL of the GMOIL distribution is given by On the other hand, the th k moment of reversed residual life (RRL) of a random variable X is defined as follows Again, we employ the binomial expansion and pdf in Eq. (21) in Eq. (31), then the th k moment of RRL of the GMOIL will be For k=1 in Eq. (32), we obtain the mean of RRL or the mean waiting time of the GMOIL distribution, which represents the waiting time elapsed since the failure of an item on condition that this failure had occurred.

Stochastic ordering
Let X and Y are independent random variables with cdfs FX and FY respectively, then according to Shaked et al. [Shaked and Shanthikumar (1994)], X is said to be less than Y if the following ordering holds; We have the following chain of implications among the various partial orderings mentioned above:

Parameter estimation
In view of the cost and time constraints, censoring is used in the statistical analysis of reliability characteristics for a system or device even with a loss in efficiency. There are several types of censoring schemes which are employed in life-testing and reliability studies. Two types of censoring are generally recognized, Type-I censoring (TIC) and Type-II censoring (TIIC). In TIC scheme, the experiment continues until a pre-assigned time τ, and failures that occur after τ are not observed. In contrast, in TIIC scheme the experiment decides to terminate after a pre-assigned number of failures observed, say r, r≤n. In this section, the point and approximate confidence intervals (CIs) estimators of the GMOIL model parameters, under TIC and TIIC schemes, are obtained using maximum likelihood (ML) method.

ML estimators based on TIC
,..., , n X X X be the observed TIC sample of size r whose life time's has the GMOIL distribution with Eq. (9) are placed on a life test and the test is terminated at specified time τ before all n items have failed. The log-likelihood function, based on TIC, of vector of parameters κ is given by: where, also, for simplicity we x . Hence the partial derivatives of the log-likelihood function with respect to , b α and θ components of the score vector can be obtained as follows where, (1 ) (1 ) , The ML estimators of the model parameters are determined by solving the Eqs. (36)-(38) after setting them with zeros. These equations cannot be solved analytically and statistical software can be used to solve them numerically via iterative technique.
The ML estimators of the model parameters are determined by solving Eqs. (40)-(42) after setting them with zeros. These equations cannot be solved analytically and statistical software can be used to solve them numerically via iterative technique. Note that, for r=n, we obtain the ML estimators of the model parameters in case of complete sample.

Approximate confidence intervals
In this subsection, approximate CIs of the model parameters for the GMOIL distribution are obtained.
The 3  α θ are respectively, given by: Here, 2 Z ν is the upper 2 ν th percentile of the standard normal distribution and var (.)'s denote the diagonal elements of 1 ( ) I κ − corresponding to the model parameters.
• Numerical outcomes of the previous measures are listed in Tabs. 2 to 5 based on TIC, while Tabs. 6 to 9 contain the numerical results in case of complete and TIIC. From these tables we conclude the following • Bias, MSE and AL of all parameters decrease as the sample size increases.
• As the value of τ increases, the bias, MSE and AL of all parameters decrease.
• As the value of r increases, the bias, MSE and AL of all parameters decrease.
• At α=2, θ=1.2 and as the value of b increases, the bias, MSE and AL of all parameters increase. • The AL of the CIs increases as the confidence levels increase from 90% to 95%.
Statistics measures like; minus log-likelihood (-log L), Kolmogorov-Smirnov (KS) test statistic, Akaike information criterion (AIC), corrected AIC (CAIC), Bayesian information criterion (BIC) and Hannan-Quinn information criterion (HQIC) are obtained. These measures are applied to test the superiority of the GMOIL distribution in comparison to some other models.

Conclusions
A new three-parameter extended form of the inverse Lindley distribution related to Marshall-Olkin-G class is proposed. The new distribution is named as the generalized Marshall-Olkin inverse Lindley distribution. Some main properties of the new model are given. Estimation of the model parameters is approached by maximum likelihood method for complete and censored samples. Point and approximate confidence interval estimators of the model parameters are obtained. Simulation study is designed to evaluate the performance of the estimates. Two applications explain that the proposed distribution provides consistently better fits than the other competitive models.