Reliability Properties of the NDL Family of Discrete Distributions with Its Inference

: The n atural discrete Lindley (NDL) distribution is an intuitive idea that uses discrete ana-logs to well-known continuous distributions rather than using any of the published discretization techniques. The NDL is a flexible extension of both the geometric and the negative binomial distributions. In the present article, we further investigate new results of value in the areas of both theoretical and applied reliability. To be specific, several closure properties of the NDL are proved. Among the results, sufficient conditions that maintain the preservation properties under useful partial orderings, convolution, and random sum of random variables are introduced. Eight different methods of estimation, including the maximum likelihood, least squares, weighted least squares, Cramér–von Mises, the maximum product of spacing, Anderson–Darling, right-tail Anderson– Darling, and percentiles, have been used to estimate the parameter of interest. The performance of these estimators has been evaluated through extensive simulation. We have also demonstrated two applications of NDL in modeling real-life problems, including count data. It is worth noting that almost all the methods have resulted in very satisfactory estimates on both simulated and real-world data.


Introduction
Interests in discrete failure data came relatively late in comparison to its continuous analog. The subject matter has, to some extent, been neglected. It was only briefly mentioned by [1]. For earlier works on discrete lifetime distributions, see [2][3][4][5]. In the last few decades, many papers have appeared in the statistical literature on the discretization of continuous distributions. The most recent discrete distributions include the discrete analogs of the continuous Burr and Pareto distributions [6], discrete analog of the continuous inverse Weibull distribution [7], and discrete analog of the generalized exponential distribution [8]. These three distributions have at least two parameters each and have not yet received any applications. Further, the moments of the three distributions are expressed in terms of either non-standard special functions or infinite sums.
In spite of all the available discrete models, there is still a great need to create more flexible discrete distributions to model several types of real data in many applied areas, such as social sciences, economics, biometrics, and reliability studies, to model different types of count data.
Recently, Al-Babtain et al. [9] introduced the natural discrete Lindley (NDL) distribution, using a mixture of geometric and negative binomial distributions. However, the authors saw that there was still a lot of room for introducing new results relating the NDL to both theoretical and applied reliability, which has motivated the authors to further study the NDL distribution.
Further details about the NDL distribution can be explored in [9]. For example, Figure 1 and Figure 2 display possible shapes for the PMF and HR function of the NDL distribution for some values of , to show that the NDL distribution is always uni-modal for all values of ∈ (0, 1), whereas its HR function is always increasing in .  In the current article, several additional theoretical reliability properties, as well as useful partial orderings, are introduced. Besides, different methods for estimating the involved parameter are explored, and their results compared.
Among the basic results, it has been shown that the hazard rate ordering of members of the NDL family is preserved under a common contamination. This is quite useful in the case where the systems are operating under random common environments. Important results are derived covering the preservation of the sums of random variables under the hazard rate, likelihood ratio, and the reversed hazard rate. Such results are quite useful in the reliability practicing. More importantly, it is shown that the life lengths of two series systems composed of ordered components of the NDL family preserve the hazard rate order. A basic result gives sufficient conditions for the preservation of the D-MRL property under the assumption of log-concavity of an added contamination. Other, similar results consider the D-MRL case. Finally, an interesting application to renewal processes, which is very helpful in "replacement studies", is also presented.
In particular, we give sufficient conditions that maintain the preservation properties under useful partial orderings, convolution, and random sum of random variables. Preservation under common random effects of the surrounding environment is also established. Moreover, interesting applications to weighted distributions and length-biased models have been carefully investigated. Finally, different methods have been used to estimate the parameter of the NDL, and the efficiencies of these estimation methods have been compared.
The rest of the paper is organized as follows. For completeness, several reliability closure properties and useful partial ordering comparisons are established in Section 2. In Section 3, a discrete renewal process application is presented. Section 4 is devoted to inferences about the involved parameter. In Section 5, we conduct a detailed simulation study to explore the behavior of the proposed estimators. In Section 6, the validity of estimation methods is checked empirically using two real biological datasets. Finally, conclusions and future work are given in Section 7.

Preliminaries
This section is devoted to presenting definitions, notation, and basic facts used throughout the paper. We use increasing (decreasing) in place of nondecreasing (nonincreasing).
The following two lemmas pave the road for introducing our new results.
Lemma 1 shows that the NDL family is ordered by different values of the parameter according to the HR order. For a proof, see Corollary 2 in [9].
The next lemma shows that the NDL has the increasing failure rate (IFR) property.
Proof. See Theorem 1 in [9]. □ Let be ( ), = 1,2. Let be a contaminated independent of , s. The following theorem shows that the HR ordering is preserved under an added contamination.
Definition 2. The discrete rv is said to be smaller than in weak likelihood ratio (WLR) or-

Definition 3. The mean residual lifetime (MRL) of the NDL distribution is given by
where ( ) is the HR function of the NDL distribution.  The following chains of implication hold [12].

Closure under Hazard and Reversed Hazard Orders
The following result shows that convolutions of members from the NDL family is preserved under the reserved HR ordering. Below, we prove that weighted NDL distributions are (under mild conditions on the weights) preserved in the reversed HRs.
Proof. Observe that the HR function, , of is given by where is the HR function of . Similarly, the HR function of, , of is given by where is the HR function of . Now, appealing to Lemma 1, it follows that Next, using Theorem 1.B.2. in [13] and the monotonicity assumption of , we get that Combining this inequality with ≥ , the result follows. □ Next, we compare the life of two series systems composed of NDL components in the HR order. Proof. If suffices to prove the theorem for = . Suppose , , and have HRs , , and , respectively. If is not hard to see that has the HR + while has the HR + .
By assumption ( ) + ( ) ≥ ( ) + ( ) for all ≥ 0, that is, ≤ . □ Theorem 5 shows that sums of members of the NDL family could be compared in the LR ordering. Proof. According to Lemma 2, and each have a log-concave PMF. Since the convolution of rvs with log-concave PMFs has a log-concave PMF, it is enough to show that if , and are independent rvs such that ≤ and has a log-concave PMF, then + ≤ + . Let , , = , and denote the PMFs of the indicated rvs. Then, The assumption ≤ means that ( ), as a function of and ∈ 1,2 , is . Now, the log-concavity of means that ( − ), as a function of and , is .
Therefore, by the basic assumption formula ( [14], P. 17), we see that ( ) is in ∈ 1,2 and . That is The following theorem proves that the NDL family is preserved under the D-MRL ordering, as well as finite mixtures of members of the family under a well-known discrete ordering of the weights.

Theorem 6. Let
, and be non-negative discrete rvs, and is independent of both and and also let has a PMF . Then, for ≤ and is log-concave implies that + ≤ + .

Proof.
We have to show that Or equivalently, Next, using the well-known composition formula ( [14], p. 17), the left side of (1) is equal to The conclusion then follows if we note that the first determinant is non-negative since is log-concave, and that the second determinant is non-negative since ≤ . □
Note that could be thought of as a joint contamination when measuring and .

Corollary 2. If
, and follow NDL distributions with parameters , , and , respectively, such that ≤ , ≤ , is independent of and is independent of , then the following statement holds: Proof. The following chain of inequalities establish the result The following theorem compares the distribution ( ) for a , and ( ) for a .
Proof. One can follow similar arguments to those used in Theorem 3.2 of [15]. □

Discrete Renewal Process Application
Let ( ), = 0,1,2, … and ( ), = 0,1,2, … denote renewal processes having inter-arrival distributions and , respectively. Proof. The proof follows by mimicking the elegant proofs of Lemma 8.5.5 and Theorem 8.6.4 of [16], and the fact that where and are two sequences of iid rvs having and as their respective CDFs.
A version of arguments used to prove Corollary 3.16 and Theorem 3.17 in Chapter 6 of [17] can be used to show that the following are valid. Assume In the next section, we shall estimate the parameter of interest using eight different methods of estimations. Then, the section is concluded by a simulation study to compare the obtained results.

Estimation Methods
In this section, we estimate the parameter of the NDL distribution, using eight methods of estimation. The methods used include the maximum likelihood estimator (MLE), least squares estimator (LSE), weighted least squares estimator (WLSE), Cramér-Von Mises estimator (CME), maximum product of spacing estimator (MPSE), Anderson-Darling estimator (ADE), right-tail Anderson-Darling estimator (RTADE), and percentiles estimator (PCE).

Maximum Likelihood Estimator
Now, we estimate the parameter of the NDL distribution using the MLE. The loglikelihood function of the NDL distribution has the form ℓ( | ) ∝ 2 log( ) − log(1 + ) + ̅ log(1 − ).
The MLE of follows by solving ℓ( | ) = 0, that is After some algebra, the MLE of , is given by the following compact formula: Al-Babtain et al. [9] showed that the MLE and moment estimator of the parameter of the NDL distribution have the same estimator in closed form. They also derived a formula to calculate the bias-correction (BI-C) as follows:

Least Squares and Weighted Least Squares Estimators
Consider the order statistics of a random sample from the NDL distribution denoted by : , : , ⋯ , : . The LSE of follows by minimizing (2) The WLSE of can be derived by minimizing the following equation with respect to Further, the WLSE of can also be calculated by solving the following non-linear equation in which ( : | ) is derived in (2).

Cramér-Von Mises Estimator
The CVME can be derived as the difference between the estimates of the CDF and the empirical CDF [18]. The CVME of can be obtained by minimizing the following equation with respect to ( ) = 1 12 The CVME of can also calculated by solving the following equation

Maximum Product of Spacing Estimator
The MPSE is a good alternative to the MLE [19,20]. Consider the uniform spacings for a random sample from the NDL distribution The MPSE of follows also, by solving the following equation:

Anderson-Darling and Right-Tail Anderson-Darling Estimators
The ADE is a type of MDE. The ADE of the parameter can be derived by minimizing

Percentiles Estimator
The percentiles approach is introduced by [21,22]. Consider the unbiased estimator of ( : | ) given by = /( + 1). Then, the PCE of the NDL parameter is derived by minimizing the following eqution with respect to : where ( ) denotes the negative branch of the Lambert function that is known as the product-log function in the Mathematica software© [23].

Simulation Results
Now, we present the results of a simulation study to explore the behavior of the pro-  Tables 1-6. One can note, from Tables 1-6, that the estimates of the parameter of the NDL distribution are entirely good; that is, these estimates are quite reliable and very close to the true values for all values of , showing small biases, MSEs, and MREs for all considered values of . For all values of , the MSEs, AABs, and MREs decrease as sample size increases, hence the eight estimators show the consistency property. We conclude that the MLE, LSE, WLSE, CME, MPSE, ADE, RTADE, and PCE perform very well in estimating the parameter of the NDL distribution. Further, the performance of these methods will be checked empirically in the next section.

Estimation Methods Based on Real Biological Data
To check the validity of the methods used in this paper to estimate the parameter of the NDL distribution, two different datasets are taken from [9]. One dataset consists of 47 observations representing the daily deaths in Egypt due to COVID-19 infections in the time interval from 8 March to 30 April, 2020. The other dataset, measuring remission times in weeks, consists of 20 leukemia patients who were randomly assigned to a certain treatment [25]. Both datasets have been used to illustrate the flexibility of the NDL distribution in modeling similar datasets in estimating the parameter . For this purpose, eight methods of estimation have been applied to both datasets.
The estimates of the parameter and goodness-of-fit statistics such as Cramér-von Mises (W), Anderson-Darling (A), and Kolmogorov-Smirnov (KS) statistics, with their associated p-value (KS p-value) are reported in Table 7. Probability-probability (PP) plots for COVID-19 and remission times data for the eight estimates are displayed in Figure 3 and Figure 4, respectively. It is shown, from Table 7 and Figure 3 and Figure 4, that all estimators perform very well.

Conclusions and Future Work
The natural discrete Lindley (NDL) distribution has been published as an application of a natural discretization method. However, the published paper did not show its possible applications in significant areas of statistics, like reliability applications. Thus, the main object of this paper has been to widen the usefulness of the NDL distribution through further study of several closure properties under different reliability properties, including the conditions leading to an IFR, as well as hazard rate ordering, reversed hazard rate ordering, and associated results based on these orders.
In addition to the basic results, we also show that the hazard rate ordering of members of the NDL family is preserved under a common contamination. This is quite useful in cases where the systems are operating under random common environments. Important results are derived covering the preservation of the sums of random variables under the hazard rate, likelihood ratio, and reversed hazard rate. Such results are quite useful in the reliability practicing. Furthermore, it is shown that the life lengths of two series systems composed of ordered components of the NDL family preserve the hazard rate order. A basic result gives sufficient conditions for the preservation of the D-MRL property under the assumption of log-concavity of an added contamination. Other, similar results consider the D-MRL case. Finally, an interesting application to renewal processes, which is very helpful in "replacement studies", is also presented.
The paper is concluded with eight different estimation methods to estimate the parameter , and a comparative study has been conducted on their results to explore the behavior of the eight proposed estimators. Two real datasets have been used to explore the behavior of these estimators for estimating the NDL parameter empirically. Each of the methods used has demonstrated acceptable results.
For a possible direction of future studies, the work in this paper can be augmented with the binomial distribution to generate a new model to handle the relationships between the number of particles entering and leaving an attenuator, where several interesting results of value in applications can be established. Compounding results based on adding the life lengths of a random number of components following the NDL distribution could be applied to several insurance problems.
Author Contributions: All the authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement:
The datasets used in this paper are provided within the main body of the manuscript.