Poisson XLindley Distribution for Count Data: Statistical and Reliability Properties with Estimation Techniques and Inference

In this study, a new one-parameter count distribution is proposed by combining Poisson and XLindley distributions. Some of its statistical and reliability properties including order statistics, hazard rate function, reversed hazard rate function, mode, factorial moments, probability generating function, moment generating function, index of dispersion, Shannon entropy, Mills ratio, mean residual life function, and associated measures are investigated. All these properties can be expressed in explicit forms. It is found that the new probability mass function can be utilized to model positively skewed data with leptokurtic shape. Moreover, the new discrete distribution is considered a proper tool to model equi- and over-dispersed phenomena with increasing hazard rate function. The distribution parameter is estimated by different six estimation approaches, and the behavior of these methods is explored using the Monte Carlo simulation. Finally, two applications to real life are presented herein to illustrate the flexibility of the new model.


Introduction
Researchers obtain a multitude of probability distributions for analyzing the various forms of data sets from diverse sectors, such as health, transportation, engineering, astronomy, and agriculture. Various well-known approaches are used to introduce new probability distributions. Some famous approaches, such as compounding technique and T-X family, give a very effective way to generalize a common parametric family of distributions to fit data sets and those classical distributions do not sufficiently fit. In some practical fields, count data may be generated/observed, and to model such data, discrete probability distributions were proposed based on different approaches such as survival discretization, Poisson mixture, and compound models. For example, Greenwood and Yule [1] compound Poisson and negative binomial distributions by considering the rate parameter in the Poisson distribution. Mahmoudi and Zakerzadeh [2] extended the Poisson-Lindley distribution and revealed that their generalized distribution is more flexible in evaluating count data. Zamani and Ismail [3] introduced a novel compound distribution by combining a negative binomial distribution with a one-parameter Lindley distribution that provides a better fit for count data. Rashid [4] introduced a count data model that combines the negative binomial and Kumaraswamy distributions and used it for modeling biological data sets. Some more discrete distributions are Poisson-Ishita distribution by Hassan et al. [5]; Poisson-Ailamujia distribution by Hassan et al. [6]; Poisson Xgamma distribution by Para et al. [7]; Poisson quasi-Lindley distribution by Grine and Zeghdoudi [8]; discrete Gompertz-G family by Eliwa et al. [9]; discrete extension to three-parameter Lindley model by Eliwa et al. [10]; two-parameter exponentiated discrete Lindley distribution by El-Morshedy et al. [11]; Eliwa and El-Morshedy [12]; discrete Burr-Hatke distribution by El-Morshedy et al. [13]; discrete Weibull Marshall-Olkin family by Gillariose et al. [14]; and discrete Ramos-Louzada model by Eldeeb et al. [15].
e XLindley (XL) distribution was introduced for the analysis of lifetime data (see [16]). Let X be a random variable following the XL distribution with the probability density function: Since there is a need for a more flexible model for modeling statistical data, in this study, we proposed a flexible discrete distribution by compounding Poisson and XL distributions. e proposed model is named the "Poisson-XL" distribution. e reported distribution strength lies in the capacity to describe equi-and over-dispersed data.
Furthermore, it can be used as a suitable statistical tool to model positively skewed data with leptokurtic shape. One more advantage to Poisson-XL model is that its statistical and reliability characterization can be expressed in closed forms, which make this model have multi-benefits in regression and time-series analysis.
e study is organized as follows. Section 2 is devoted to the derivation of Poisson-XL distribution and its shape analysis. Some statistical properties are derived in Section 3. Some reliability measures are derived in Section 4. e parameter is estimated in Section 5. Section 6 is based on the applications of the proposed distribution. In the end, we concluded this study in Section 7.

Synthesis of the Poisson-XL Model
If X|λ follows Poisson(λ) where λ is itself a random variable following XL distribution with parameter α, then determining the distribution that results from marginalizing over λ will be known as a compound of Poisson distribution with that of XL distribution, which is denoted by the PXL model.

Theorem 1.
e probability mass function of a compound of PXL distribution is given as follows: Proof. e probability mass function of a compound of Poisson(λ) with XL(α) can be formulated as follows: en: where x � 0, 1, 2, . . . , and α > 0. Figure 1 shows the probability mass function (PMF) plots of the proposed distribution for various values of parameter α. According to Figure 1, it is noted that the PMF can be either unimodal or decreasing-shaped. Further, it can be utilized as a probability tool to discuss right-skewed data. e corresponding cumulative distribution function (CDF) to equation (2) can be expressed as follows: where x � 0, 1, 2, . . . , and α > 0. Let x 1: n , x 2: n , x 3: n, . . . , x n: n be the order statistics of a random sample from the PXL distribution. e cumulative distribution function of ith order statistics for an integer value of x is given as follows: Computational Intelligence and Neuroscience ] m+j represent the CDF of the exponentiated PXL distribution with power m + j.
e corresponding PMF to equation (2) is given as follows: where f i (x; α, m + j) represents the PMF of the exponentiated PXL distribution with power parameter m + j. us, the pth moments of X i: n can be written as follows: and when d/dx(P(X � λ, α)) � 0, the solution is as follows: For α, x > 0, the mode is a unique critical point, in which P(X � λ, α) is maximum and P(X � λ, α) is concave, but if x < 0 the density function is decreasing of x.

Factorial Moments.
e rth factorial moment around the origin of the PXL distribution can be obtained as follows: where X (r) � X(X − 1)(X − 2) . . . (X − r + 1), and then: and assuming y � x − r, we get the following:

Probability Generating Function (PGF)
Theorem 2. If X has PXL(X; α), then PGF G x (Z) can be formulated as follows: Proof.
e PGF can be obtained as follows: Computational Intelligence and Neuroscience

Moment Generating Function (MGF)
Theorem 3. If X has PXL(X; α), then the MGF can be expressed as follows:

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Proof.
e moments around the origin can be obtained as follows: en: e first four ordinary moments of X are as follows: whereas the first four moments around the mean of X are as follows: α 11 + 18α 10 + 127α 9 + 515α 8 + 1395α 7 + 2692α 6 +3747α 5 + 3678α 4 + 2430α 3 + 1010α 2 + 240α + 24 Based on the rth moments, the index of dispersion index (DI) can be expressed as follows: Further, the skewness and kurtosis can be derived in closed forms, where: e summary measure, mean, variance, moments, and DI are presented in Table 1. e PXL model can be used to model equi-and over-dispersed data. e plots of coefficient of skewness and kurtosis are shown in Figure 2. e skewness and kurtosis monotonically increase for higher values of α. Moreover, the PXL model can be used as a probability tool for modeling positively skewed data with leptokurtic shape.

Shannon Entropy.
e Shannon entropy is a measurable physical property that is most associated with a state of disorder, randomness, or uncertainty. e term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the microscopic description of nature in statistical physics, and to the principles of information theory. e Shannon entropy of the random variable X can be expressed as follows: and then: Computational Intelligence and Neuroscience  For more details around HurwitzLerchPhi (0, 1, 0) function "Lerch transcendent," see https://mathworld.wolfram.com/ LerchTranscendent.html. Some entropy values of PXL distribution in terms of the parameter (α) are presented in Table 2. It is noticed that the Shannon entropy shows a monotonically decreasing pattern and it proceeds to zero when α increased.

Reversed (Hazard) Rate Function and Mills
Ratio. e corresponding survival function (SF) to equation (5) can be expressed as follows: e hazard rate function (HRF) of the random variable X can be defined as h(x) � P(X � x; α)/S(x − 1; α). en, the HRF of the PXL distribution can be formulated as follows: It is easy to see that the limiting behavior HRF at the upper limit is lim x⟶∞ h(x) � (α/1 + α). As a result, the parameter α may be regarded as a strict upper bound on the HRF, which is a key feature of lifetime probability distributions. Few discrete distributions contain parameters that can be readily interpretable in terms of failure rate functions. e geometric distribution is an exception, although in this instance the HRF is constant. In Proposition 1, we proved that the PXL distribution always allows for increasing failure rates.

Proposition 1.
e HRF of the PXL distribution is increasing.
Proof. According to Glaser (1980) and from the PMF of the PXL distribution: and it follows that: ∀x, α > 0, implying that h(x) is increasing. Figure 3 illustrates some plots of the PXL model based on various values of the model parameter. e reverse hazard rate of the PXL distribution is as follows: whereas the second rate of failure and Mills ratio can be expressed, respectively, as follows: where N 0 � 0, 1, 2, . . . , w { } and 0 < w < ∞. Let X have the PXL random variable, and then, the MRL is defined as follows: Computational Intelligence and Neuroscience and after simple algebra steps, we get the MRL in an explicit form as follows:

Various Estimation Techniques
is section is based on parameter estimation of the PXL distribution using different estimation methods. e considered methods are maximum likelihood, moment, Anderson darling, Cramér-von Mises, ordinary least squares, and weighted least squares.

Maximum-Likelihood Estimation (MLE)
. Suppose x � (x 1 , x 2 , x 3 , . . . , x n ) be a random sample of size "n" from the PXL distribution. en, the log-likelihood (L) function is given as follows: Partially differentiating with respect to α, we get the following: Since we cannot get a close form to equation (15), a numerical procedure should be used to solve this equation numerically to get the maximum-likelihood estimator.

Method of Moment Estimation (MOME).
Based on the MOME approach for estimating the parameter, the sample and population means should be derived. So, to get the estimator of the PXL model, the solution of the following nonlinear equation provides the estimate of α, where:

Anderson-Darling Estimation (ADE).
e ADE is based on the difference in empirical and fitted CDF. e ADE of α follows by minimizing: with respect to α.
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Cramér-von Mises Estimation (CVME).
e CVME is based on the difference between empirical and fitted CDF. e CVME of α follows by minimizing: with respect to α.

Ordinary Least-Squares and Weighted Least-Squares
Estimation. Let X i: n be the ith order statistics in a sample of size n. We adopt lower cases for sample values. It is well known that: us, the least-squares estimate (LSE) of α, say α, can be derived by minimizing: with respect to α. e weighted least-squares estimate (WLSE) of α, say α, can be determined by minimizing: with respect to α.

Simulation
To assess the accuracy of the six estimators described previously, we conducted a comprehensive simulation study. We used the PXL distribution to generate samples with n � 25, 50, 100, 200, and 500 and then calculated the average values (AVEs) of the MLE, MOME, LSE, WLSE, CVME, and ADE to get the mean square errors (MSEs), average absolute biases (ABBs), and mean relative errors (MREs) for α � 0.3, 0.5, 1.0, and 1.5. e ABBs, MREs, and MSEs are given as follows: We ran the simulation 5000 times to derive these metrics from the prior values for all estimation methods. e findings in Tables 3-6 were obtained using the R software's optim-CG function. e findings show that as the sample size n increased, the AVEs became closer to the real values of α. Furthermore, when n increases, the ABBs, MREs, and MSEs for all estimators decreased.

Applications
In this section, the flexibility of the PXL distribution is proposed based on two distinctive real data sets. e first data set is the biological experiment data on the European corn borer [17], which is shown in Table 7. e investigator counts the number of borers per hill of corn in an experiment conducted randomly on 8 hills in 15 replications. e mean, variance, and index of dispersion values of X are 1.4833, 3.193, and 2.1526, respectively. Since distribution is over-dispersed, we can use PXL distribution.
e second data set shows the number of mammalian cytogenetic dosimetry lesions produced by streptogramin (NSC-45383) exposure in rabbit lymphoblasts of − 70 3 bc g/ kg [18]. e second data set is shown in Table 8. e mean, variance, and index of dispersion values of X are 0.54, 0.8312, and 1.5392, respectively.
Some competitive models such as the discrete Bilal (DB) by Altun et al. [19]; discrete Pareto (DPr) by Krishna and Pundir [20]; discrete Rayleigh (DR) by Roy [21]; discrete Burr-Hatke (DBH) by El-Morshedy et al. (2020); discrete inverted Topp-Leone (DITL) by Eldeeb et al. [22]; Poisson-Ailamujia (PA) by Hassan et al. [6]; and Poisson (Poi) distributions are used herein. To obtain the best model to analyze data sets I and II, some criteria should be used such as Akaike information criterion (AIC) and Bayesian information criterion (BIC) as well as − L as indicators of the       Computational Intelligence and Neuroscience relative quality of statistical models for the given set of data. ese criteria assess the quality of each model with the other models given a set of data models. Moreover, the chi-square (χ 2 ) test is used with its corresponding p value where the estimated probabilities under the null hypothesis are as follows: e estimated expected frequencies are obtained as e i � nα i . e results of the chi-square test are reported in Tables 7 and 8. us, we cannot reject the null hypothesis at the 5% level of significance and the PXL distribution is a good fit for these data sets.
For data set I, the PXL and PA work quite well for analyzing data set I, but the PXL is the best, and Figure 4 supports our empirical results, which are listed in Table 7. For data set II, the PXL, DBH, and DITL work quite well for analyzing data set II, but the PXL is the best, and Figure 5 supports our empirical results, which are reported in Table 8. Since one of the major aims of this study is to get the best estimators for the data sets I and II, several estimation techniques have been derived for this purpose. Tables 9 and 10 report the different estimators     It is noted that MLE and MOME approaches work quite well in analyzing data set I, but the MLE method is the best for these data, whereas data set II can be discussed via the MLE and MOME techniques, but the MLE is the best.

Conclusion
In this study, a new one-parameter Poisson-XLindley (PXL) distribution has been proposed for modeling count data. Some distributional properties are derived and studied in detail. It was found that the properties of the PXL can be expressed in closed forms, which make it a proposer probability tool to establish regression and time-series model for discussing different types of data sets in various fields. e new probability mass function can be utilized to model positively skewed data with leptokurtic shape. Moreover, the PXL model can be used to model equi-and over-dispersed phenomena with increasing hazard rate function. Different estimation approaches have been used to estimate the model parameter.
e behavior of these methods has been explored using the Monte Carlo simulation. Finally, two applications to real life have been discussed to illustrate the flexibility of the new discrete model.

Conflicts of Interest
e authors declare that they have no conflicts of interest.