An Extended log-Lindley-G Family: Properties and Experiments in Repairable Data

: In this article, two new families of distributions are proposed: the generalized log-Lindley-G (GLL-G) and its counterpart, the GLL*-G. These families can be justiﬁed by their relation to the log-Lindley model, an important assumption for describing social and economic phenomena. Speciﬁc GLL models are introduced and studied. We show that the GLL density is rewritten as a two-member linear combination of the exponentiated G-densities and that, consequently, many of its mathematical properties arise directly, such as moment-based expressions. A maximum likelihood estimation procedure for the GLL parameters is provided and the behavior of the resulting estimates is evaluated by Monte Carlo experiments. An application to repairable data is made. The results argue for the use of the exponential law as the basis for the GLL-G family.


Introduction
Providing flexible and accurate distributions is sought in a variety of contexts, including public health, biomedical studies, and reliability (for a fuller discussion, see [1]). Lindley [2] pioneered the one-parameter Lindley distribution, to which several extensions have been developed [3]. Abouammoh and Kayid [4] developed a general form for Lindeybased extended models. Al-Babtain et al. [5] proposed a new version for a discrete analog of the continuous Lindley law. Beyond extensions of the Lindley distribution, improved statistical inference tools have been derived from it [6][7][8][9].
Recently, Gómez-Déniz et al. [10] have proposed the log-Lindley (LL) distribution with probability density function (pdf) According to Zakerzadeh and Dolati [11], this distribution may be also developed from the generalized Lindley law and has simple moment-based expressions and important reliability properties. The new model has been successfully applied in the current context on which the LL cdf is often used to misinterpret the prime principle (Gómez-Déniz et al. [10]). Our proposal is an alternative to the family proposed by Jones [12], who furnished a flexible family of models from Beta random variables. This family of distributions is able to model symmetric and skewed data sets with varying tail weights. Further, Zografos and Balakrishnan [13], Ristic and Balakrishnan [14] and Amini et al. [15] suggested a family of distributions spawned from the gamma random variables. In a similar vein, we present two new families of distributions generated from log-Lindley random variables through the following definition.
Recently, Alzaatreh et al. [16] introduced the T-X family-constructed from a random variable T ∈ [a, b] for a < b with density r(t)-having cumulative distribution function (cdf) and its associated pdf is Based on the T-X generator defined by Equation (2), we introduce two new families of continuous distributions termed generalized log-Lindley-G distributions by integrating the LL density function. Definition 1. Denote F(·) andF(·) as the cdf and the survival function of a random variable with the pdf f (·). We announce two new families of distributions having pdfs The proposed family can actually be obtained from the log-Lindley family of Gómez-Déniz et al. [10] and the T-X generator, taking into account the integrals and respectively. First, we define the density functions of r(t) in (2) as and and replacing L[G(x)] by G α (x) in (2), we have the two cdf of the generalized log-Lindley-G family, respectively, as and where G(x; ζ) and g(x; ζ) depend on a parameter vector ζ and α, λ, σ are three additional parameters. Henceforth, we will refer to these families as GLL-G, F(·), and GLL * -G, F * (·) families. The structure of the paper is as follows. In Section 2 we propose two new families of distributions arising from the LL model. Section 3 brings some special models in these families. In Section 4 we derive an extension for the density of the family GLL-G. In Section 5, we derive several mathematical properties of the GLL-G family, including, along with ordinary and incomplete moments, mean deviations and the moments of residual and reversed lifetime. Rényi, Shannon, and Mathai-Haubold entropies and order statistics are studied in Sections 6 and 7, respectively. An estimation procedure for the GLL-G parameters is given in Section 8. Sections 9 and 10 deal with the numerical results, while Section 11 contains the concluding thoughts.

The GLL-G Family
Here, we introduce two new families of distributions having cdfs (10) and (11) by incorporating two additional shape parameters, α > 0 and σ > 0, and one scale parameter, λ ≥ 0, to yield a more flexible generator. By differentiating (10) and (11), the associated pdfs are, respectively, and The hazard rate functions (hrfs) of the proposed families defined in (10) and (11) are given as respectively. Now, we offer expressions for the quantile functions related to (10) and (11). Based on the work of Jodrá and Jiménez-Gamero [17], it holds from F(X) = U and F * (X * ) = U that, respectively, and where W(·) is the secondary branch Lambert W function. The last identities furnish random number generators (RNGs) for distributions F * (·) and F(·). Some notable existing members of the families (12) and (13) are introduced in Table 1.

Family Survival Function (Parameter Restriction) Density Reference
Lindley distribution The inverse power Lindley distribution The weighted log-Lindley distribution Lindley-exponential distribution Transformed gamma distribution

Special GLL Generalized Laws
Some particular models from the GLL-G family (12) are given as functions of three known distributions. In particular, these are the exponential, the Maxwell-Boltzmann (MB), and the Lévy distributions, each of which has densities of

The GLL-MB Distribution
Setting the MB pdf and cdf as input for (13) and (14), the following functions hold: where Erf(·) is the error function. This case is typified by X ∼ GLL-MB(α, λ, σ, β).

The GLL-L Distribution
Taking the Lévy pdf and cdf as input for (13) and (14), one follows where Erfc(·) is the complementary error function. This case is addressed by X ∼ GLL-L(α, λ, σ, β).

Expansion for the Density Function
Considerable mathematical developments of the proposed families can be derived by means of linear combinations of pdfs of the exponentiated-G ("Exp-G" for short) family. In this section, we find a helpful expansion for the pdf (12) as a double linear combination of Exp-G densities. Then, we find an expansion for f * (x; α, λ, σ, ζ).
For a σ > 0 real non-integer, we have the power series expansion where, for any real number, the binomial cofficient is defined. Applying the expansion (17) to (12), we have If σ is an integer, the indicator i in the preceding sum stops at σ − 1. Further, the following expansion holds for |z| < 1: By inserting (19) in (18), where Finally, the pdf (20) can be represented as a double mixture of infinite linear combinations of the Exp-G densities where ω i = v i α(i+1) , ω i,j = η i,j /α(i + j + 2) and H b (x) = G b (x) and h b (x) = bg(x)G(x) b−1 denote the Exp-G cdf and pdf with power parameter b, respectively. Therefore, various mathematical tools of the novel distribution family can be obtained from the Exp-G model, such as different kinds of moments.

Mathematical Properties
We will develop a number of mathematical features of the GLL-G distribution family (12) in this section: moments, moment generating function (mgf), mean deviations and (reversed) residual lifetime moments.
These measures have several applications in practice. For instance, in economics, the total proportion of a company's accumulated revenue can be defined in terms of including ordinary and incomplete moments. Lorenz and Bonferroni curves also involve these moments and are of great applicability in social and political phenomena.

Moments
Let Y α(i+1) and Y α(i+j+2) be two distributed exp-G random variables having densities h α(i+1) (x) and h α(i+j+2) (x), respectively. Then, the nth moment of X, say µ n , follows from (21) as Nadarajah and Kotz [26] presented expressions for the moments of various exp-G distributions, which can be utilized to achieve µ n . Setting n = 1 in (22) gives the mean of X.
Other expressions for E[X n ] can be derived from (22) in terms of the G quantile function for ρ > 0. Note that for Y ρ following the Exp-G law, is the baseline quantile function.

Incomplete Moments
The nth incomplete moment of X is given by where m n,ρ (y) = G(y;ζ) This integral can be evaluated for the majority of the G-distributions.

Generating Function
Then, we contribute with two formulations for the mgf M x (t). The early one may be obtained from (21) as where M α(i+1) (t) and M α(i+j+2) (t) are the mgfs of Y α(i+1) and Y α(i+j+2) , respectively. Hence, M X (t) can be derived from the Exp-G generating function. in terms of the G quantile function, another expression for M X (t) can be found from (24) as: for ρ > 0, where M ρ (t) is the mgf of Y ρ . This integral can be numerically evaluated for many of the similar distributions.

Mean Deviations
The mean deviations around the mean (δ 1 ) and for the median (δ 2 ) of X can be revealed as respectively, where M is the median of X (obtained defining U = 1/2 in (15) and (16)) and m 1 (z) is the first incomplete moment achieved from (23) at n = 1.

Moments of Residual and Reversed Lifetimes
The residual and reversed residual lifetime random variables play an important role in reliability theory. The moments of such variables are extensively used in actuarial sciences and analysis of risks. The moments of residual and reversed lifetimes of X can be formulated as (for t ≥ 0) and respectively, where F * (·), f * (·) and E(X n ) are given by (10), (12) and (22), respectively, and m n (t) is the nth incomplete moment which can be calculated from (23).

Entropy
Information theory is a branch of mathematics concerned with the quantification of information between communication channels [27]. One of its most important measures is entropy, which describes the degree of disorder in a stochastic system. Now, we tackle three common entropy measures, developed by Shannon [28], Rényi [29] and Mathai and Haubold [30], defined as and respectively, where γ > 0, γ = 1, δ = 1 and δ < 2. The following relation is satisfied: Several works have focused on the derivation of these measures for specific distributions [31][32][33]. These measures are derived for one of the proposed families in what follows.

Rényi Entropy
From Equation (12), we can write Applying the power series (17) to (32), we have Expanding the power of (− log[1 − G α (x)]) as: where (for j ≥ 0), p j,0 = 1 and (for k = 1, 2, . . .), Additionally, if we use the binomial expansion of the last equation and insert it in the equation above related to the expansion of power series in (33), we have the follow expression: Then, the entropy reduces to the following: The Rényi entropy can be used to characterize symmetric distributions [34].

Mathai-Haubold Entropy
Since after some algebraic manipulations, (35) reduces to Applying all the above results, we get Among other things, Mathai-Haubold entropy can be strongly associated with the theory of record values [35].

Order Statistics
Order statistics has been used for a variety of applied (reliability analysis, censored sample analysis, . . . ) and theoretical (including characterization of probability distributions and goodness-of-fit tests, robust statistical estimation, entropy estimation, . . . ) contexts. In the following, we derive expressions for the cdf and the density of the GLL family.
The pdf of the ith order statistic, say, X i:n , for a random sample from GLL-G (12), is expressed by Based on (10), we have whereω m,k,r,h = (−1) k+m+h i + j − 1 h αk m k r σ r λ r−h (1 + λσ) k and a s = (s + 1) −1 .
It is noticeable that the density of the GLL-G order statistics consists in an Exp-G density mixture. As a result, for Exp-G distributions, the mgf and moments of GLL-G order statistics derive instantly from linear combinations of those values.

Estimation of the Parameters
The maximum likelihood estimates (MLEs) for GLL-G parameters are discussed here. Let x 1 , x 2 , . . . , x n be observed values from the GLL-G distributions given by (12) with parameter vector θ = (α, λ, σ, ζ T ) T , where ζ = [ζ 1 , . . . , ζ p ] T is the p-dimensional parameter vector. Then, the log-likelihood function l = l(θ) for θ is constructed by The score vector members are determined by and ∂l ∂ζ k and

Simulation Study
Now we are able to perform some simulation studies to confirm the asymptotic properties of the MLEs. The simulation study was made following the Monte Carlo approach. For this purpose, we use the RNG developed in Section 2. We use bias (difference between the average of the MLEs and the true parameter value) and mean squared error (MSE) as performance measures. The best performances are associated with the smallest values of bias and MSE.
In terms of simulation configuration, we assumed:   (15) works well. It should be emphasized that the acceptance-rejection simulation method is a good alternative when the latter (derived from the inversion method) fails.  Table 2 shows the result of the simulation study. In general, estimating λ and σ was more difficult than estimating α and β. However, when the sample size was increased, both the MSE and the bias decreased.

Applications
We compare the performance of the generalized log-Lindley-G family having three different baselines as input: exponential, Lévy and Maxwell distributions. The database we used can be found in Hamedani et al (2017) [37], and it represents the time between failures for repairable items. Figure 2 provides both fits of cdfs and pdfs to the data for the considered models. It is possible to note by visual inspection that the GLL-Exp distribution outperformed the remaining models. Table 3 exhibits MLEs and their associated standard errors (SEs). It is noticeable that these values were well defined in all cases; i.e., asymptotic confidence intervals did not include zero. Table 4 exhibits the results of four goodness-of-fit (GoF) measurements that were employed in this comparison study: Akaike information criterion (AIC), corrected AIC (AICc), Kolmogorov-Smirnov (K-S) and Bayesian information criterion (BIC) statistics. Confirming previous qualitative discussions, the values indicate that the GLL-Exp distribution had the greatest performance.

Conclusions
We have established two new families of distributions in this paper named generalized log-Lindley families, which stem from the log-Lindley family. We have obtained several of their mathematical properties, such as expansions for density and cumulative distribution functions, entropies, moment generating functions, entropies and stochastic ordering. Maximum likelihood (ML) estimation procedures have been proposed as well. We have seen through simulation that the ML estimators had behavior aligned with it as expected asymptotically. Finally an application was made in order to illustrate the new family. Results favored the GLL-Exp law relative to the GLL-MB and -L distributions.