A New Scale-Invariant Lindley Extension Distribution and Its Applications

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 the α -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.


Introduction
e Lindley distribution introduced by Lindley [1] has been attracted much interest in recent years. e probability density function (PDF) and the cumulative distribution function (CDF) of the Lindley are, respectively, as follows: Clearly the Lindley distribution is a mixture of two gamma distributions G(1, θ), gamma with shape parameter 1 and scale θ and G(2, θ) with weights θ/1 + θ and 1/1 + θ, respectively.
One distribution family f(x) is said to be scale invariant if f(x) 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. e aim of this paper is to introduce one scale invariant extension of the Lindley distribution.
In Sections 2 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.

New Lindley Extension
We introduce one extended Lindley (EL) distribution with parameters k > 0, m > 0, k ≠ m, δ > 0, and θ > 0, namely, EL(k, m; δ, θ), with the PDF as follows: It is a mixture of two gamma distributions G(m, θ) and G(k, θ) with weights Γ(m)/Γ(m) + δ and δ/Γ(m) + 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.
For m � 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.
It is straightforward to show that the CDF of EL(k, m; δ, θ) is in which Γ(x, α) � ∞ x t α− 1 e − t dt is the upper incomplete gamma function. Let X ∼ EL(k, m; δ, θ), then the r th moment is It is clear that when k > 1 and m > 1, E(X r ) > 1/θ r . e moment-generating function can also be simplified to Applying Glaser's technique, this failure rate function may exhibit increasing, bathtub-shaped, or increasing then bathtub-shaped forms (see Glaser [24] or eorem 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.

Proposition 1.
e MRL function of X ∼ EL(k, m; δ, θ) is Proof. e MRL can be expressed by e numerator of (8) can be simplified as However, Similarly, en, by (8)- (11), the result follows. e α-QRL is defined to be When X ∼ EL(k, 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 Figure 3, 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 the α-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. e 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]).

Power Extended Lindley
Now, we introduce the power Lindley extended (PEL) distribution with parameters k > 0, m > 0, k ≠ m, δ > 0, θ > 0, and c > 0, namely, PEL(k, m; δ, θ, c), with the PDF as follows: For m � 2, k � 1, θ � δ, and c � 1, it is the Lindley model. On the other hand, when δ � 0 and c � 1, it reduces to the gamma distribution with PDF as follows: If Y ∼ EL(k, m; δ, θ), then X � Y 1/c ∼ PEL(k, m; δ, θ, c). is statement is useful for simulating random samples of PEL(k, m; δ, θ, c) and studying its properties. Figure 4 shows the PDF for some various values of the parameters and indicates that the PDF accommodates decreasing, unimodal, and biomodal forms.
It is straightforward to show that the CDF of PEL(k, m; δ, θ, c) is It is clear that when k > 1 and m > 1, E(X r ) > θ − (r/c) . By the fact that e tX � ∞ j�0 (tX) j /j! and applying (16), the moment-generating function can be simplified to

Dynamic Measures.
Let X ∼ PEL(k, m; δ, θ, c) with the reliability function R(x), then the failure rate function is is failure rate function will accommodate increasing, bathtub, and increasing then bathtub-shaped forms since it is a power model of the baseline model EL(k, m; δ, θ). e form of the failure rate of the power model depends on the baseline model and the value of c. For example, if the failure rate function of the baseline model be increasing and c > 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 c > 1, then the failure rate function of the power model will be bathtub-shaped or increasing and so on. Figure 5 draws the failure rate function for some values of parameters and confirms that it accommodates increasing, bathtub, and increasing then bathtub-shaped forms graphically.
Proof. e MRL can be expressed by e numerator of (20) can be simplified as However, Similarly, en, by (20)-(23), the result follows. e α-QRL is defined to be 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 the α-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.

Parameter Estimation
Let X i , i � 1, 2, . . . , n represent an independent and identically distributed (iid) sample from EL(k, 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: For m ≥ 1 and k ≥ 1, the right side of these equations is greater than 1/θ and 1/θ 2 , respectively. So, one lower bound for θ is max((1/X), (1/X 2 ) 0.5 ).
e log-likelihood function for (δ, θ) when k and m are known is  Density function Failure rate function e MLE can be computed by directly maximizing (26) or solving the log-likelihood equations (27) and (28) simultaneously.
e Fisher information matrix in one sample of size n about (δ, θ) is where l � ln f(X). Let K − 1 be the inverse of the information matrix, then asymptotically, which can be applied for constructing confidence intervals for the parameters or in hypothesis testing. When k 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.

Parameter Estimation of the Power Extended Lindley.
Let X i , i � 1, 2, . . . , n represent an iid sample from PEL(k, m; δ, θ, c), and k and m are known. Via the moment method, by (16), the triple (δ, θ, c) can be estimated by finding their solution in the following system of equations: e log-likelihood function for (δ, θ, c) when k and m are known is l(δ, θ, c; X) � n ln(c) +(c − 1) n i�1 ln X i + n ln(θ) − n ln(Γ(m) + δ) e log-likelihood equations are e MLE can be computed by maximizing (32) directly or solving the log-likelihood equations (33)-(35) simultaneously.
e Fisher information matrix in one sample of size n about (δ, θ, c) is where l � ln f(X). Assume that K − 1 be the inverse of the information matrix, then Var(δ, θ, c) ≈ n − 1 K − 1 . Furthermore, let X i , i � 1, 2, . . . , n represent an iid sample from asymptotically, which can be applied for constructing confidence intervals for the parameters or in hypothesis testing. Similarly, when k and/or m are unknown, we can estimate (δ, θ, c) 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.

Right-Censored Data.
Assume that we have one iid random sample X i , i � 1, 2, . . . , n from the EL (PEL) which is exposed to right censorship. en, X i is censored from right by a censoring random variable C i , if C i ≤ X i , so the only information about event time is that it is greater than censoring time C i . us, the observations consist of T i � min(X i , C i ) and d i , where d i � 1, when the event is not censored, X i ≤ C i , and d i � 0, when the event is censored, X i > C i . With the censored data in hand, the log-likelihood function is denoted by l c (δ, θ; t, d)(l c (δ, θ, c; t, d)) and equals to 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, θ)(δ, θ, c) to find the MLE and estimate its variance by reverse of the Fisher information matrix.

Simulation Study
e proposed distribution is a mixture of two gamma distribution, so we can follow the below steps to generate random samples of size n of it.
(1) Simulate one random instance from binomial distribution with parameters n and δ/Γ(m) + δ. Let n 1 be the generated instance and n 2 � n − n 1 . (2) Simulate one random sample of size n 1 from G(k, θ) and one random sample of size n 2 from G(m, θ). en, merge these two sample to provide one random sample from EL(k, m, δ, θ).
where R is the reliability function of the extended Lindley. e result of this investigation for the EL model is abstracted in Table 1. 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. en, three pairs (B δ , B θ ), (AB δ , AB θ ), and (MSE δ , MSE θ ) in which and B θ , AB θ , and MSE θ 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:

Applications
In this section, four data sets, (see Tables 3-6) have been considered. For every data set, the EL(k, m; δ, θ) and PEL(k, m; δ, θ, c) have been fitted, for all pairs (k, m) where k � 1, 2, . . . , 10 and m � 1, 2, . . . , 10. en, 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. e 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 shows the number of cycles to failure for 25 specimens of yarn, considered in Shanker [28]. e 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. e second data set (see Table 4) 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. e PEL gives the lowest value for the K-S, CVM, and AD statistics. So, PEL outperforms other models from this point of view. e 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. Table 5 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. e estimated PDF and the scaled TTT plot have been plotted in Figures 12 and 13, respectively. Table 6 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.

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. e models are not very complicated but accommodate different and useful forms of hazard rate function (MRL or median residual life functions). e 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.

Data Availability
e lifetime data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest.