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

Authors: Muhammad Ahsan-ul-Haq; Afrah Al-Bossly; Mahmoud El-Morshedy; Mohamed S. Eliwa
Journal: Computational Intelligence and Neuroscience
 (2022)

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## Abstract

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.


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## Body

## 1. 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].The 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:(1)fx=α22+α+xe−αx1+α2;x>0,α>0.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. The proposed model is named the “Poisson-XL” distribution. The 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.The study is organized as follows. Section2 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. The 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.


## 2. Synthesis of the Poisson-XL Model

IfX|λ follows Poissonλ 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.
The probability mass function of a compound of PXL distribution is given as follows:(2)Px=PPXLDX=x,α=α2x+α2+31+α1+α4+x;x=0,1,2,…,α>0.Proof.
The probability mass function of a compound ofPoissonλ with XLα can be formulated as follows:(3)PX=λ,α=∫0∞p1x|λf1x,λdλ.Then:(4)PX=λ,α=∫0∞e−λλxx!α22+α+λe−αλ1+α2dλ,=α21+α2x!∫0∞λx2+α+λe−λα+1dλ,=α2x+α2+31+α1+α4+x,where x=0,1,2,…,andα>0. Figure 1 shows the probability mass function (PMF) plots of the proposed distribution for various values of parameter α.
According to Figure1, 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. The corresponding cumulative distribution function (CDF) to equation (2) can be expressed as follows:(5)FX=x,α=∑y=0xα2y+α2+31+α1+α4+y=1−11+α4+x1+αx+4+α3+α.where x=0,1,2,…,andα>0. Let x1:n,x2:n,x3:n,…,xn:n be the order statistics of a random sample from the PXL distribution. The cumulative distribution function of ith order statistics for an integer value of x is given as follows:(6)Fi:nx;α=∑m=innmFix;αm1−Fix;αn−m=∑m=in∑j=0n−mΨkn,mFix;αm+j=∑m=in∑j=0n−mΨkn,mFix;α,m+j, whereΨkn,m=−1jnmn−mj and Fix;α,m+j=1−1+αx+4+α3+α/1+α4+xm+j represent the CDF of the exponentiated PXL distribution with power m+j. The corresponding PMF to equation (2) is given as follows:(7)fi:nx;α=Fi:nx;α−Fi:nx−1;α=∑m=in∑j=0n−mΨkn,mfix;α,m+j,where fix;α,m+j represents the PMF of the exponentiated PXL distribution with power parameter m+j. Thus, the pth moments of Xi:n can be written as follows:(8)EXi:np=∑x=0∞∑m=in∑j=0n−mΨkn,mxpfix;α,m+j.Figure 1
PMF plot of the PXL distribution.


## 3. Statistical Properties

### 3.1. Mode

To get and study the mode of the PXL model with its characterization, we should derive the first and second derivatives to the PMF with respect tox, where:(9)ddxPX=λ,α=α2−α2log1+αx+α2+3α+11+α4+x,d2dx2PX=λ,α=−2α2log1+α+log1+α2x+α2+3α+11+α4+x,and when d/dxPX=λ,α=0, the solution is as follows:(10)x^=1log1+α−α2+3α+3,d2dx2PX=λ,αx=x^=−2α2log1+α1+α1/log1+α−α2−3α+1<0.Forα,x^>0, the mode is a unique critical point, in which PX=λ,α is maximum and PX=λ,α is concave, but if x^<0 the density function is decreasing of x.


### 3.2. Factorial Moments

Therth factorial moment around the origin of the PXL distribution can be obtained as follows:(11)μr′=μr′=EEXr|λ=∫0∞∑x=0∞xre−λλxx!α22+α+λe−αλ1+α2dλ,where Xr=XX−1X−2…X−r+1, and then:(12)μr′=α21+α2∫0∞λr∑x=r∞e−λλx−rx−r!2+α+λe−αλdλ,and assuming y=x−r, we get the following:(13)μr′=α21+α2∫0∞λr∑y=0∞e−λλyy!2+α+λe−αλdλ,μr′=α21+α22+α∫0∞λre−αλdλ+∫0∞λr+1e−αλdλ=αα+2+r+11+α2.


### 3.3. Probability Generating Function (PGF)

Theorem 2.
IfX has PXLX;α, then PGF GxZ can be formulated as follows:(14)GxZ=α21+α3z+1+α−zα2+31+α1+α−z2;α>0.Proof.
The PGF can be obtained as follows:(15)GxZ=∑x=0∞Zxpx=∑x=0∞Zxα2x+α2+31+α1+α4+x=α21+α4∑x=0∞Zxx+α2+31+α1+αx=α21+α4∑x=0∞xZx1+αx+α2+31+α∑x=0∞Zx1+αx=α21+α4z1+α1+α1+α−z2+α2+31+α1+α1+α−z=α21+α3z1+α−z2+α2+31+α1+α−z.


### 3.4. Moment Generating Function (MGF)

Theorem 3.
IfX has PXLX;α, then the MGF can be expressed as follows:(16)Mxt=α21+α3et+1+α−etα2+31+α1+α−et2;α>0.Proof.
The moments around the origin can be obtained as follows:(17)Mxt=∑x=0∞etxPx=∑x=0∞etxα2x+α2+31+α1+α4+x=α21+α4∑x=0∞xetx1+αx+α2+31+α∑x=0∞etx1+αx=α21+α4et1+α1+α1+α−et2+α2+31+α1+α1+α−et.Then:(18)Mxt=α21+α3et1+α−et2+α2+31+α1+α−et.
The first four ordinary moments ofX are as follows:(19)μ1′=α2+2α+2α1+α2,μ2′=α3+4α2+6α+6α21+α2,μ3′=α4+8α3+20α2+30α+24,α31+α2μ4′=α5+16α4+66α3+138α2+192α+120α41+α2,whereas the first four moments around the mean of X are as follows:(20)μ1=α2+2α+2α1+α2,μ2=α5+5α4+11α3+14α2+10α+2α21+α4,μ3=α8+9α7+36α6+87α5+141α4+152α3+98α2+30α+4α31+α6,μ4=1α41+α8α11+18α10+127α9+515α8+1395α7+2692α6+3747α5+3678α4+2430α3+1010α2+240α+24.
Based on the rth moments, the index of dispersion index (DI) can be expressed as follows:(21)DI=VarXEX=α5+5α4+11α3+14α2+10α+2α4+2α3+3α2+2α+2.
Further, the skewness and kurtosis can be derived in closed forms, where:(22)S=μ3−3μ2μ1+2μ13μ21.5,K=μ4−4μ3μ1+6μ2μ12−3μ14μ22.
The summary measure, mean, variance, moments, and DI are presented in Table1. The PXL model can be used to model equi- and over-dispersed data.
The plots of coefficient of skewness and kurtosis are shown in Figure2. The 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.Table 1
Moments and DI of the PXL distribution.
αMeanVarianceDIEX2EX3EX40.118.265215.2511.785548.8422486.01162376.90.28.472256.1386.6261127.922679.371386.30.52.88899.65433.341918.000160.221847.30.71.92295.13172.66878.829258.293502.481.01.25002.68752.15004.250020.750133.251.50.77331.34861.74391.94676.924432.5482.00.55560.85801.54441.16673.388913.0002.50.43270.61771.42770.80492.02746.72163.00.35420.47871.35170.60421.36814.05793.50.29980.38931.29850.47920.99872.71114.00.26000.32741.25920.39500.77001.94384.50.22960.28221.22910.33490.61781.46715.00.20560.24771.20530.29000.51091.15137.00.14510.16611.14500.18720.28970.5602100.10080.11101.10080.12120.16800.2825500.02000.02041.02000.02080.02250.02591000.01000.01011.01000.01020.01060.0114Figure 2
Skewness and kurtosis of the PXL distribution.


### 3.5. Shannon Entropy

The Shannon entropy is a measurable physical property that is most associated with a state of disorder, randomness, or uncertainty. The 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. The Shannon entropy of the random variableX can be expressed as follows:(23)Hx=−∑x=0∞PX=x;αlogPX=x;α,and then:(24)Hx=−∑x=0∞α2x+α2+31+α1+α4+xlogα2+logx+α2+3α+3−4+xlog1+α=−logα2−α21+α4∑x=0∞x+α2+31+αlogx+α2+31+α1+αx+log1+αα21+α4∑x=0∞x+α2+31+α4+x1+αx=−logα2+α2HurwitzLerchPhi0,1,01/1+α,−1,α2+31+α1+α4+2+10α+23α2+28α3+17α4+4α5log1+αα1+α4.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.Table 2
Shannon entropy of PXL distribution.
αHxαHxαHx0.13.882823.00.778196.50.458580.23.170563.50.702017.00.435220.52.213364.00.641447.50.414441.01.545384.50.591938.00.395821.51.214625.00.550568.50.379012.01.013845.50.515391000.056112.50.877566.00.485065000.01443


## 3.1. Mode

To get and study the mode of the PXL model with its characterization, we should derive the first and second derivatives to the PMF with respect tox, where:(9)ddxPX=λ,α=α2−α2log1+αx+α2+3α+11+α4+x,d2dx2PX=λ,α=−2α2log1+α+log1+α2x+α2+3α+11+α4+x,and when d/dxPX=λ,α=0, the solution is as follows:(10)x^=1log1+α−α2+3α+3,d2dx2PX=λ,αx=x^=−2α2log1+α1+α1/log1+α−α2−3α+1<0.Forα,x^>0, the mode is a unique critical point, in which PX=λ,α is maximum and PX=λ,α is concave, but if x^<0 the density function is decreasing of x.


## 3.2. Factorial Moments

Therth factorial moment around the origin of the PXL distribution can be obtained as follows:(11)μr′=μr′=EEXr|λ=∫0∞∑x=0∞xre−λλxx!α22+α+λe−αλ1+α2dλ,where Xr=XX−1X−2…X−r+1, and then:(12)μr′=α21+α2∫0∞λr∑x=r∞e−λλx−rx−r!2+α+λe−αλdλ,and assuming y=x−r, we get the following:(13)μr′=α21+α2∫0∞λr∑y=0∞e−λλyy!2+α+λe−αλdλ,μr′=α21+α22+α∫0∞λre−αλdλ+∫0∞λr+1e−αλdλ=αα+2+r+11+α2.


## 3.3. Probability Generating Function (PGF)

Theorem 2.
IfX has PXLX;α, then PGF GxZ can be formulated as follows:(14)GxZ=α21+α3z+1+α−zα2+31+α1+α−z2;α>0.Proof.
The PGF can be obtained as follows:(15)GxZ=∑x=0∞Zxpx=∑x=0∞Zxα2x+α2+31+α1+α4+x=α21+α4∑x=0∞Zxx+α2+31+α1+αx=α21+α4∑x=0∞xZx1+αx+α2+31+α∑x=0∞Zx1+αx=α21+α4z1+α1+α1+α−z2+α2+31+α1+α1+α−z=α21+α3z1+α−z2+α2+31+α1+α−z.


## 3.4. Moment Generating Function (MGF)

Theorem 3.
IfX has PXLX;α, then the MGF can be expressed as follows:(16)Mxt=α21+α3et+1+α−etα2+31+α1+α−et2;α>0.Proof.
The moments around the origin can be obtained as follows:(17)Mxt=∑x=0∞etxPx=∑x=0∞etxα2x+α2+31+α1+α4+x=α21+α4∑x=0∞xetx1+αx+α2+31+α∑x=0∞etx1+αx=α21+α4et1+α1+α1+α−et2+α2+31+α1+α1+α−et.Then:(18)Mxt=α21+α3et1+α−et2+α2+31+α1+α−et.
The first four ordinary moments ofX are as follows:(19)μ1′=α2+2α+2α1+α2,μ2′=α3+4α2+6α+6α21+α2,μ3′=α4+8α3+20α2+30α+24,α31+α2μ4′=α5+16α4+66α3+138α2+192α+120α41+α2,whereas the first four moments around the mean of X are as follows:(20)μ1=α2+2α+2α1+α2,μ2=α5+5α4+11α3+14α2+10α+2α21+α4,μ3=α8+9α7+36α6+87α5+141α4+152α3+98α2+30α+4α31+α6,μ4=1α41+α8α11+18α10+127α9+515α8+1395α7+2692α6+3747α5+3678α4+2430α3+1010α2+240α+24.
Based on the rth moments, the index of dispersion index (DI) can be expressed as follows:(21)DI=VarXEX=α5+5α4+11α3+14α2+10α+2α4+2α3+3α2+2α+2.
Further, the skewness and kurtosis can be derived in closed forms, where:(22)S=μ3−3μ2μ1+2μ13μ21.5,K=μ4−4μ3μ1+6μ2μ12−3μ14μ22.
The summary measure, mean, variance, moments, and DI are presented in Table1. The PXL model can be used to model equi- and over-dispersed data.
The plots of coefficient of skewness and kurtosis are shown in Figure2. The 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.Table 1
Moments and DI of the PXL distribution.
αMeanVarianceDIEX2EX3EX40.118.265215.2511.785548.8422486.01162376.90.28.472256.1386.6261127.922679.371386.30.52.88899.65433.341918.000160.221847.30.71.92295.13172.66878.829258.293502.481.01.25002.68752.15004.250020.750133.251.50.77331.34861.74391.94676.924432.5482.00.55560.85801.54441.16673.388913.0002.50.43270.61771.42770.80492.02746.72163.00.35420.47871.35170.60421.36814.05793.50.29980.38931.29850.47920.99872.71114.00.26000.32741.25920.39500.77001.94384.50.22960.28221.22910.33490.61781.46715.00.20560.24771.20530.29000.51091.15137.00.14510.16611.14500.18720.28970.5602100.10080.11101.10080.12120.16800.2825500.02000.02041.02000.02080.02250.02591000.01000.01011.01000.01020.01060.0114Figure 2
Skewness and kurtosis of the PXL distribution.


## 3.5. Shannon Entropy

The Shannon entropy is a measurable physical property that is most associated with a state of disorder, randomness, or uncertainty. The 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. The Shannon entropy of the random variableX can be expressed as follows:(23)Hx=−∑x=0∞PX=x;αlogPX=x;α,and then:(24)Hx=−∑x=0∞α2x+α2+31+α1+α4+xlogα2+logx+α2+3α+3−4+xlog1+α=−logα2−α21+α4∑x=0∞x+α2+31+αlogx+α2+31+α1+αx+log1+αα21+α4∑x=0∞x+α2+31+α4+x1+αx=−logα2+α2HurwitzLerchPhi0,1,01/1+α,−1,α2+31+α1+α4+2+10α+23α2+28α3+17α4+4α5log1+αα1+α4.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.Table 2
Shannon entropy of PXL distribution.
αHxαHxαHx0.13.882823.00.778196.50.458580.23.170563.50.702017.00.435220.52.213364.00.641447.50.414441.01.545384.50.591938.00.395821.51.214625.00.550568.50.379012.01.013845.50.515391000.056112.50.877566.00.485065000.01443


## 4. Reliability Characteristics of the PXL Distribution

### 4.1. Reversed (Hazard) Rate Function and Mills Ratio

The corresponding survival function (SF) to equation (5) can be expressed as follows:(25)Sx;α=11+α3+x1+αx+4+α3+α;x=0,1,2,3,….The hazard rate function (HRF) of the random variableX can be defined as hx=PX=x;α/Sx−1;α. Then, the HRF of the PXL distribution can be formulated as follows:(26)hx=α2x+α2+31+α1+α1+αx+α2+31+α.It is easy to see that the limiting behavior HRF at the upper limit islimx⟶∞hx=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. The 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.
The HRF of the PXL distribution is increasing.Proof.
According to Glaser (1980) and from the PMF of the PXL distribution:(27)ηx=−P′xPx=−1+log1+αx+α2+31+αx+α2+31+α,and it follows that:(28)η′x=1x+α2+31+α2>0,∀x,α>0, implying that hx is increasing. Figure 3 illustrates some plots of the PXL model based on various values of the model parameter.
The reverse hazard rate of the PXL distribution is as follows:(29)r∗x=PxFx=α2x+α2+31+α1+α4+x−1+αx+4+α3+α,whereas the second rate of failure and Mills ratio can be expressed, respectively, as follows:(30)r∗∗x=logSxSx+1=log1+α1+αx+α2+3+3α1+αx+α2+4+3α,M=Sxpx=1+1+αx+4+α3+αα2x+α2+31+α.Figure 3
HRF plots of the PXL distribution.


### 4.2. Mean Residual Life

For the random variableX, the mean residual life or the mean remaining lifetime (MRL) is the expected remaining life of X−t, given that the item has survived to time i. Thus, the MRL concept can be used effectively in stochastic ageing and dependence for reliability. The unconditional mean of the distribution, EX, is a special case of mean residual life for i=0. For a discrete random variable, the MRL function is defined as follows:(31)εi=EX−i|X≥i=11−Fi−1,α∑j=i+1w1−Fj−1,α;i∈ℕ0,where ℕ0=0,1,2,…,w and 0<w<∞. Let X have the PXL random variable, and then, the MRL is defined as follows:(32)εi=1+α3+i1+αi+3+α3+α∑j=i+1w1+αj+3+α3+α1+α3+j=1+αi1+αi+3+α3+α∑j=i+1w1+αj+3+α3+α1+αj,and after simple algebra steps, we get the MRL in an explicit form as follows:(33)εi=1+αi1+αi+3+α3+α1+α3+α3+α∑j=i+1w11+αj+α∑j=i+1wj1+αj=11+αi+3+α3+α1+α3+α3+α1α+αi+1+αα=1+αiα+4+α4+αα1+αi+3+α3+α.


## 4.1. Reversed (Hazard) Rate Function and Mills Ratio

The corresponding survival function (SF) to equation (5) can be expressed as follows:(25)Sx;α=11+α3+x1+αx+4+α3+α;x=0,1,2,3,….The hazard rate function (HRF) of the random variableX can be defined as hx=PX=x;α/Sx−1;α. Then, the HRF of the PXL distribution can be formulated as follows:(26)hx=α2x+α2+31+α1+α1+αx+α2+31+α.It is easy to see that the limiting behavior HRF at the upper limit islimx⟶∞hx=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. The 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.
The HRF of the PXL distribution is increasing.Proof.
According to Glaser (1980) and from the PMF of the PXL distribution:(27)ηx=−P′xPx=−1+log1+αx+α2+31+αx+α2+31+α,and it follows that:(28)η′x=1x+α2+31+α2>0,∀x,α>0, implying that hx is increasing. Figure 3 illustrates some plots of the PXL model based on various values of the model parameter.
The reverse hazard rate of the PXL distribution is as follows:(29)r∗x=PxFx=α2x+α2+31+α1+α4+x−1+αx+4+α3+α,whereas the second rate of failure and Mills ratio can be expressed, respectively, as follows:(30)r∗∗x=logSxSx+1=log1+α1+αx+α2+3+3α1+αx+α2+4+3α,M=Sxpx=1+1+αx+4+α3+αα2x+α2+31+α.Figure 3
HRF plots of the PXL distribution.


## 4.2. Mean Residual Life

For the random variableX, the mean residual life or the mean remaining lifetime (MRL) is the expected remaining life of X−t, given that the item has survived to time i. Thus, the MRL concept can be used effectively in stochastic ageing and dependence for reliability. The unconditional mean of the distribution, EX, is a special case of mean residual life for i=0. For a discrete random variable, the MRL function is defined as follows:(31)εi=EX−i|X≥i=11−Fi−1,α∑j=i+1w1−Fj−1,α;i∈ℕ0,where ℕ0=0,1,2,…,w and 0<w<∞. Let X have the PXL random variable, and then, the MRL is defined as follows:(32)εi=1+α3+i1+αi+3+α3+α∑j=i+1w1+αj+3+α3+α1+α3+j=1+αi1+αi+3+α3+α∑j=i+1w1+αj+3+α3+α1+αj,and after simple algebra steps, we get the MRL in an explicit form as follows:(33)εi=1+αi1+αi+3+α3+α1+α3+α3+α∑j=i+1w11+αj+α∑j=i+1wj1+αj=11+αi+3+α3+α1+α3+α3+α1α+αi+1+αα=1+αiα+4+α4+αα1+αi+3+α3+α.


## 5. Various Estimation Techniques

This section is based on parameter estimation of the PXL distribution using different estimation methods. The considered methods are maximum likelihood, moment, Anderson darling, Cramér–von Mises, ordinary least squares, and weighted least squares.


### 5.1. Maximum-Likelihood Estimation (MLE)

Supposex=x1,x2,x3,…,xn be a random sample of size “n” from the PXL distribution. Then, the log-likelihood L function is given as follows:(34)L=2nlogα+∑i=1nlogxi+α2+31+α−∑i=1n4+xilog1+α.Partially differentiating with respect toα, we get the following:(35)∂L∂α=2nα+∑i=1n2α+3xi+α2+31+α−∑i=1n4+xi1+α.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.


### 5.2. 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 ofα, where:(36)x¯=α2+2α+2α1+α2.


### 5.3. Anderson–Darling Estimation (ADE)

The ADE is based on the difference in empirical and fitted CDF. The ADE ofα follows by minimizing:(37)ADEα=−n−1n∑i=1n2i−1log1−11+α4+i1+αxi:n+4+α3+α+log11+α4+i1+αxi:n+4+α3+α,with respect to α.


### 5.4. Cramér–von Mises Estimation (CVME)

The CVME is based on the difference between empirical and fitted CDF. The CVME ofα follows by minimizing:(38)CVMEα=112n+∑i=1n1−11+α4+i1+αxi:n+4+α3+α−2i−12n2,with respect to α.


### 5.5. Ordinary Least-Squares and Weighted Least-Squares Estimation

LetXi:n be the ith order statistics in a sample of size n. We adopt lower cases for sample values. It is well known that:(39)EFXi:n=i1+nandVFXi:n=in−i+1n+12n+2.Thus, the least-squares estimate (LSE) ofα, say α^, can be derived by minimizing:(40)LSEα=∑i=1n1−11+α4+i1+αxi:n+4+α3+α−in+12,with respect to α. The weighted least-squares estimate (WLSE) of α, say α^, can be determined by minimizing:(41)WLSEα=∑i=1nn+12n+2in−i+11−11+α4+i1+αxi:n+4+α3+α−in+12,with respect to α.


## 5.1. Maximum-Likelihood Estimation (MLE)

Supposex=x1,x2,x3,…,xn be a random sample of size “n” from the PXL distribution. Then, the log-likelihood L function is given as follows:(34)L=2nlogα+∑i=1nlogxi+α2+31+α−∑i=1n4+xilog1+α.Partially differentiating with respect toα, we get the following:(35)∂L∂α=2nα+∑i=1n2α+3xi+α2+31+α−∑i=1n4+xi1+α.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.


## 5.2. 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 ofα, where:(36)x¯=α2+2α+2α1+α2.


## 5.3. Anderson–Darling Estimation (ADE)

The ADE is based on the difference in empirical and fitted CDF. The ADE ofα follows by minimizing:(37)ADEα=−n−1n∑i=1n2i−1log1−11+α4+i1+αxi:n+4+α3+α+log11+α4+i1+αxi:n+4+α3+α,with respect to α.


## 5.4. Cramér–von Mises Estimation (CVME)

The CVME is based on the difference between empirical and fitted CDF. The CVME ofα follows by minimizing:(38)CVMEα=112n+∑i=1n1−11+α4+i1+αxi:n+4+α3+α−2i−12n2,with respect to α.


## 5.5. Ordinary Least-Squares and Weighted Least-Squares Estimation

LetXi:n be the ith order statistics in a sample of size n. We adopt lower cases for sample values. It is well known that:(39)EFXi:n=i1+nandVFXi:n=in−i+1n+12n+2.Thus, the least-squares estimate (LSE) ofα, say α^, can be derived by minimizing:(40)LSEα=∑i=1n1−11+α4+i1+αxi:n+4+α3+α−in+12,with respect to α. The weighted least-squares estimate (WLSE) of α, say α^, can be determined by minimizing:(41)WLSEα=∑i=1nn+12n+2in−i+11−11+α4+i1+αxi:n+4+α3+α−in+12,with respect to α.


## 6. 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 withn=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. The ABBs, MREs, and MSEs are given as follows:(42)ABBs=∑i=1Nα^−αN,MREs=∑i=1Nα^−α/aN,MSE=∑i=1Nα^−α2N.We ran the simulation 5000 times to derive these metrics from the prior values for all estimation methods. The findings in Tables3–6 were obtained using the R software’s optim-CG function. The 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.Table 3
Simulation results of PXL distribution forα=0.3.
nMLEMOMEADECVMEOLSEWLSE25AVEs0.33420.33410.29070.29130.29120.2863500.32960.33070.28860.28870.28870.28191000.32790.32790.28760.28840.28810.27822000.32640.32600.28740.28760.28760.27505000.32570.32610.28750.28730.28720.271125AABs0.03420.03410.00930.00870.00880.0137500.02960.03070.01140.01130.01130.01811000.02790.02790.01240.01160.01190.02182000.02640.02600.01260.01240.01240.02505000.02570.02610.01250.01270.01280.028925MREs0.16260.04870.11680.12150.12340.1135500.12520.03830.08740.08680.08790.08921000.10410.03120.06570.06700.06630.07952000.09120.02690.05310.05360.05330.08405000.08590.02620.04450.04530.04560.096325MSEs0.00420.00410.00190.00210.00210.0017500.00230.00240.00100.00100.00110.00111000.00150.00150.00060.00060.00060.00082000.00100.00100.00040.00040.00040.00085000.00080.00080.00020.00030.00030.0009Table 4
Simulation results of PXL distribution forα=0.5.
nMLEMOMEADECVMEOLSEWLSE25AVEs0.60160.60060.46290.46140.46130.4387500.58920.59140.46180.45880.45870.42821000.58480.58600.45940.45770.45670.41642000.58180.58340.45850.45640.45630.40655000.58000.58110.45830.45660.45620.395025AABs0.10160.10060.03710.03860.03870.0613500.08920.09140.03820.04120.04130.07181000.08480.08600.04060.04230.04330.08362000.08180.08340.04150.04360.04370.09355000.08000.08110.04170.04340.04380.105025MREs0.22950.11530.12390.12440.12670.1379500.18830.09650.09950.10350.10290.14541000.17200.08740.08850.09140.09320.16722000.16390.08350.08430.08860.08850.18705000.16010.08110.08350.08680.08760.209925MSEs0.02340.02290.00560.00560.00580.0066500.01360.01420.00350.00380.00380.00631000.01000.01030.00270.00280.00290.00752000.00800.00830.00220.00240.00240.00905000.00690.00710.00190.00210.00210.0111Table 5
Simulation results of PXL distribution forα=1.0.
nMLEMOMEADECVMEOLSEWLSE25AVEs1.54691.55420.78980.77500.77170.6919501.49101.48750.78750.77030.77080.66581001.45921.46360.78570.77040.76850.64032001.44751.45040.78650.76920.76830.61805001.44081.44490.78510.76870.76810.592425AABs0.54690.55420.21020.22500.22830.3081500.49100.48750.21250.22970.22920.33421000.45920.46360.21430.22960.23150.35972000.44750.45040.21350.23080.23170.38205000.44080.44490.21490.23130.23190.407625MREs0.55290.55980.21070.22530.22850.3082500.49190.48800.21250.22970.22920.33421000.45920.46360.21430.22960.23150.35972000.44750.45040.21350.23080.23170.38205000.44080.44490.21490.23130.23190.407625MSEs0.49420.51890.05170.05740.05880.0994500.32170.31690.04900.05610.05600.11371000.24410.24990.04790.05430.05520.13032000.21610.21900.04660.05410.05450.14635000.20030.20430.04660.05380.05410.1663Table 6
Simulation results of PXL distribution forα=1.5.
n1.5MLEMOMEADECVMEOLSEWLSE25AVEs3.33893.37990.98000.95440.95490.8498503.05943.05950.98210.95350.95500.81331002.95212.94740.98060.95340.95330.78172002.89502.88000.98080.95220.95230.75405002.85582.86120.98080.95180.95200.720925AABs1.83891.87990.52000.54560.54510.6502501.55941.55950.51790.54650.54500.68671001.45211.44740.51940.54660.54670.71832001.39501.38000.51920.54780.54770.74605001.35581.36120.51920.54820.54800.779125MREs1.22711.88120.34660.36370.36340.4335501.03961.55950.34520.36430.36330.45781000.96811.44740.34630.36440.36450.47892000.93001.38000.34610.36520.36510.49735000.90391.36120.34610.36550.36530.519425MSEs6.61357.29130.27650.30360.30320.4279503.24283.28770.27130.30170.30010.47431002.40862.40460.27140.30030.30050.51732002.07922.03150.27040.30080.30070.55725001.88571.90280.26980.30080.30060.6073


## 7. Applications

In this section, the flexibility of the PXL distribution is proposed based on two distinctive real data sets. The first data set is the biological experiment data on the European corn borer [17], which is shown in Table 7. The investigator counts the number of borers per hill of corn in an experiment conducted randomly on 8 hills in 15 replications. The 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.Table 7
Goodness of fit for data set I.
XObserved frequencyExpected frequencyPXLDBPoiDPrDRDBHDITLPA04347.132.727.264.515.968.152.239.613529.239.640.420.136.222.030.433.721717.824.330.09.734.610.514.121.531110.712.514.85.621.06.07.512.2456.36.05.53.78.93.84.46.5543.72.71.62.62.72.52.83.3612.21.20.41.90.61.71.91.7721.30.50.11.50.11.31.30.8821.70.40.010.40.04.25.40.7Total120120120120120120120120120αMLE0.86610.27671.48331.11121.87430.86551.98400.6740S.E.0.08220.15980.11110.10270.08740.03850.18320.0667χ21.82279.642821.89836.24360.17925.1426.93422.7097Degree of freedom44343344p Value0.76830.0468<0.01<0.01<0.01<0.01<0.010.1394−L200.63204.68219.19220.62235.23214.05205.15201.22AIC403.26411.35440.38443.24472.45430.10412.30404.44BIC406.04414.14443.16446.02475.24432.89415.09407.23The 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]. The second data set is shown in Table 8. The mean, variance, and index of dispersion values of X are 0.54, 0.8312, and 1.5392, respectively.Table 8
Fitted PXL distribution and other competitor distributions to second data set.
XObserved frequencyExpected frequencyPXLDBPoiDPrDRDBHDITLPA0200194.6174.9174.8216.9124.8209.6196.9186.015768.594.694.443.9140.354.169.379.123024.024.025.516.232.519.919.925.2378.45.24.67.82.38.57.27.14≥64.51.30.715.20.17.96.82.5Total300300300300300300300300300αMLE2.05121.23320.54001.85180.96430.60303.71341.8519S.E.0.17630.05510.04240.11240.02980.03760.22430.1639χ23.550826.54730.43122.66396.6506.43617.44229.3468Degree of freedom22231332p Value0.1694<0.01<0.01<0.01<0.010.09220.0591<0.01−l299.31311.74314.23312.94371.12301.70302.76302.41AIC600.63625.48630.45627.88744.23605.41607.53606.83BIC604.33629.18634.16631.59747.94609.11611.23610.53Some 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 relative quality of statistical models for the given set of data. These criteria assess the quality of each model with the other models given a set of data models. Moreover, the chi-square χ2 test is used with its corresponding p value where the estimated probabilities under the null hypothesis are as follows:(43)α^i=P^X=i=α2i+α2+31+α1+α4+i;i=0,1,2,3,….The estimated expected frequencies are obtained ase^i=nα^i. The results of the chi-square test are reported in Tables 7 and 8. Thus, 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 Figure4 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 for data sets I and II based on various estimation techniques.Figure 4
Fitted PMFs of all selected models for the first data set.Figure 5
Fitted PMFs of all selected models for the second data set.Table 9
Estimation and goodness of fit for data set I.
Method↓Statistics⟶αχ2p ValueADE0.6011614.496290.02456CVME0.5956215.045850.01990OLSE0.5954215.045850.01990WLSE0.5090628.80345<0.001MOME0.867471.8395270.76524Table 10
Estimation and goodness of fit for data set II.
Method↓Statistics⟶αχ2p ValueADE0.9025096.36338<0.001CVME0.87576104.0459<0.001OLSE0.87573104.0550<0.001WLSE0.67663189.2819<0.001MOME2.050813.5583620.16666It 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.


## 8. 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. The 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. The 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.



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