# The New Novel Discrete Distribution with Application on COVID-19 Mortality Numbers in Kingdom of Saudi Arabia and Latvia

Authors: M. Nagy; Ehab M. Almetwally; Ahmed M. Gemeay; Heba S. Mohammed; Taghreed M. Jawa; Neveen Sayed-Ahmed; Abdisalam Hassan Muse
Journal: Complexity
 (2021)

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## Abstract

This paper aims to introduce a superior discrete statistical model for the coronavirus disease 2019 (COVID-19) mortality numbers in Saudi Arabia and Latvia. We introduced an optimal and superior statistical model to provide optimal modeling for the death numbers due to the COVID-19 infections. This new statistical model possesses three parameters. This model is formulated by combining both the exponential distribution and extended odd Weibull family to formulate the discrete extended odd Weibull exponential (DEOWE) distribution. We introduced some of statistical properties for the new distribution, such as linear representation and quantile function. The maximum likelihood estimation (MLE) method is applied to estimate the unknown parameters of the DEOWE distribution. Also, we have used three datasets as an application on the COVID-19 mortality data in Saudi Arabia and Latvia. These three real data examples were used for introducing the importance of our distribution for fitting and modeling this kind of discrete data. Also, we provide a graphical plot for the data to ensure our results.


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## Body

## 1. Introduction

Modeling pandemics is significant in our life as it makes it easier for researchers to understand the behavior of the spread of each virus and its effect on humanity. Nowadays, a new virus has risen on the top of the scene, Severe Acute Respiratory Syndrome coronavirus 2 (SARS-CoV-2), which causes COVID-19. This virus attracts the interest of many researchers who tried many attempts to model daily deaths in the entire world by the effect of COVID-19 infection. As an example of these studies, Al-Babtain et al. [1] introduced a natural discrete Lindley distribution and studied the mortality numbers in Egypt from 8 March to 30 April 2020. Also, Hasab et al. [2] make a study on the COVID-19 mortality numbers by using the susceptible infected recovered (SIR) epidemic dynamics of COVID-19 pandemic to model COVID-19 infections in Egypt. Algarni et al. [3] discussed type-I half-logistic Burr XG family with application of COVID-19 data. Almetwally [4] discussed the odd Weibull inverse Topp–Leone distribution with applications to COVID-19 data. Almetwally et al. [5] discussed new distribution with applications to the COVID-19 mortality rate in two different countries. El-Morshedy et al. [6] studied a new discrete distribution, called discrete generalized Lindley, to analyze the counts of the daily COVID-19 cases in `Hong Kong and daily new deaths in Iran. Maleki et al. [7] used an autoregressive time-series model regarding the two-scale mixture normal distribution to predict the retrieved and reported COVID-19 occurrences. Nesteruk [8] forecasts the daily new COVID-19 occurrences in China by using the mathematical model SIR. Batista [9] used a logistic growth regression model to estimate the final size and its peak time of the COVID-19 epidemic. Muse et al. [10] discussed modeling the COVID-19 mortality rate with a new versatile modification of the log-logistic distribution. Liu et al. [11] presented a new statistical model called arcsine-modified Weibull distribution for modeling COVID-19 patients’ data.Afify and Mohamed [12] developed the extended odd Weibull exponential (EOWE) distribution for data modeling in many sciences such as architecture, medicine, and reliability. The EOWE distribution is a flexible model offering different density function forms such as left-skewed, symmetrical, right-skewed, and reversed-J; see, the work of Alshenawy et al. [13]. Its hazard rate function (HRF) may provide declining, constant, rising, upside-down bathtub and J-shaped hazard rates, and bathtub and modified bathtub hazard ratings are quite important in terms of durability technologies. For more details, see the work of Alshenawy et al. [13]. Generally speaking, most distributions are used to model such data and can usually take four or five parameters to achieve these hazard rates. DEOWE distribution has three parameters only, and it can be used to analyze censored data due to its easy, closed forms of its HRF and cumulative distribution function (CDF).The CDF and probability mass function (PMF) of the DEOWE distribution are given, respectively, by(1)Fx;α,β,λ=1−1+βexpλx−1α−1/β,x>0,α,β,λ>0,and(2)fx;α,β,λ=αλexpαλx1−exp−λxα−11+βexpλx−1α−1/β−1,x>0,α,β,λ>0.Why do we need discrete distributions is a question that any researcher would ask. The reason is that most current continuous distributions do not provide reliable findings for modeling the COVID-19 scenarios. The reason for all of this, as we all know, is that death counts or regular new cases display extreme dispersion.Many authors have introduced discrete distributions to overcome the deficiencies of the continuous distribution in modeling mortality numbers, such as Para and Jan [14] have introduced discrete Burr-type XII and discrete Lomax distributions. Discrete Lomax (DL) distribution is the discrete distribution which exhibits heavy tails and can be helpful in medical science and other fields, discrete Burr (DB), which is presented by Krishna and Pundir [15], discrete Lindley (DL), which is introduced by Gómez-Déniz and Calderín-Ojeda [16], discrete generalized exponential (DGEx), which is presented by Nekoukhou et al. [17], natural discrete Lindley (NDL), which is introduced by Al-Babtain et al. [1], and discrete Gompertz Exponential (DGzEx), which is presented by El-Morshedy et al. [6]. Gillariose et al. [18] introduced discrete Weibull Marshall–Olkin family of distributions with properties, characterizations, and applications. Discrete Marshall–Olkin generalized exponential distribution has been presented by Almetwally et al. [19]. Al-Babtain et al. [20] discussed the estimation of the parameters of two discrete models called discrete Poisson–Lindley and discrete Lindley distributions, with some applications.To convert a continuous distribution to a discrete one, a variety of methods are possible. A survival discretization approach is the most widely used technique for generating discrete distributions. It necessitates the existence of CDF, the existence of a continuous and nonnegative survival function, and the division of period through unit intervals. In Roy [21], the probability mass function (PMF) of a discrete distribution is described as(3)PX=x=Px≤X<x+1=Sx−Sx+1;x=0,1,2,…,where Sx=PX≥x=1−Fx;Θ, where Fx;Θ is a CDF of continuous distribution and Θ is a vector of parameters. The random variable X is said to have the discrete distribution if its CDF is given by PX<x=Fx+1;Θ. The hazard rate is given by hrx=PX=x/Sx. The reversed failure rate of discrete distribution is given as rfrx=PX=x/1−Sx.The novelty and the motivation to write this paper is to find the best statistical model which can provide the fit for COVID-19 mortality numbers in Saudi Arabia and Latvia by introducing a new discrete model, namely, the DEOWE distribution. The point estimation of the unknown parameters has been discussed by using the MLE method. Also, we make an expectation for the mortality number in each day.The remainder of this article is organized as follows. In Section2, we define DEOWE distribution. DEOWE linear representation of its PMF is obtained in Section 3, along with some of its statistical properties. The MLE method is used for parameter estimation in Section 4. In Section 5, we performed a simulation study to study the performance of the distribution relative to the true values of the parameters; also, we evaluated the relative bias (Rbias) and mean square error (MSE) of the estimation method. Two real datasets were used as three real data applications on the mortality numbers in Section 6. These three applications were used to prove that the proposed distribution provides the efficiency of the DEOWE distribution with respect to other distributions by evaluating the information criteria and the P values and chi-square values for all distributions. Finally, conclusions and the major findings are given in Section 7.


## 2. DEOWE Distribution

In this section, we introduce the DEOWE distribution, the PMF, and the CDF which are obtained. Some figures with different values of the parameters for the PMF and HRF of the distribution are represented in Figures1 and 2.Figure 1
Different shapes’ PMF of DEOWE distribution by using different values of the parameters. These figures show that the PMF can behave in different shapes.Figure 2
Different shapes’ HRF of DEOWE distribution by using different values of the parameters. These figures show that the HRF can behave in different shapes.The DEOWE distribution is obtained based on the survival discretization method. LetSx;ϑ=1−Fx;ϑ denote the survival function (S) of a baseline model with parameter vector ϑ, respectively, so the CDF of the DEOWE distribution is given by(4)Fx;α,β,λ=1−1+βeλx+1−1α−1/β,x=0,1,2,…∞.α,β,λ>0.The corresponding PMF of (4) is defined by(5)PX=x;α,β,λ=1+βeλx−1α−1/β−1+βeλx+1−1α−1/β,x=0,1,2,…∞.α,β,λ>0,where α,β,andλ are positive parameters. The random variable with PMF (5) is denoted by X∼ DEOWE α,β,λ ; the corresponding HRF of the DEOWE distribution is defined by(6)hX=x;α,β,λ=1+βeλx−1α1+βeλx+1−1α−1/β−1.


## 3. Mathematical Properties

This section of the paper introduces the linear representation of the DEOWE distribution with its quantile function.


### 3.1. Linear Representation

In this section, we made a linear representation for the PMF of the proposed distribution. We used linear representation to derive different statistical properties of the proposed model. Unfortunately, we reach a result form which does not follow any statistical model, and it is mathematically difficult to use to derive different statistical properties. In the case of the proposed distribution, we have three different cases for this linear representation.For|x|<1, we have the following expansion:(7)1+x−n=∑k=0∞−1kn+k−1kxk.For|x|>1, we have the following expansion:(8)1+x−n=∑k=0∞−1kn+k−1kx−k+n.Case 1.
Ifβeλx−1α<1 and x≠0, then we have(9)1+βeλx−1α−1/β=∑k,m,w=0∞−1k+m+w1/β+k−1kαkmβkλwm−αkww!xw,and if βeλx+1−1α<1 and x≠0, then we have(10)1+βeλx+1−1α−1/β=∑k,m,w=0∞−1k+m+w1/β+k−1kαkmβkλwm−αkww!e−λm−αkxw.
From equations (7) and (8), we have a linear representation of PMF (5) as the following:(11)PX=x;α,β,λ=∑k,m,w=0∞Φk,m,wxw,where Φk,m,w=−1k+m+w1/β+k−1kαkmβkλwm−αkw/w!1−e−λm−αk.Case 2.
. Ifβeλx−1α>1, then we have(12)1+βeλx−1α−1/β=∑k,m,w=0∞−1k+m+w1β+k−1kαk+1β+m−1m×β−k+1/βλwm+αk+1/βww!xw.
Ifβeλx+1−1α>1, then we have(13)1+βeλx+1−1α−1/β=∑k,m,w=0∞−1k+m+w1β+k−1kαk+1β+m−1m×β−k+1/βλwm+αk+1/βww!e−λm+αk+1/βxw.
From equations (9) and (10), we have a linear representation of PMF (5) as the following:(14)PX=x;α,β,λ=∑k,m,w=0∞Ψk,m,wxw,where Ψk,m,w=−1k+m+w1/β+k−1kαk+1/β+m−1mβ−k+1/βλwm+αk+1/βw/w!1−e−λm+αk+1/β.Case 3.
Ifβeλx−1α<1 and βeλx+1−1α>1, then, from equations (7) and (10), we have a linear representation of PMF (5) as the following:(15)PX=x;α,β,λ=∑k,m,w=0∞−1k+m+w1β+k−1kαkmβkλwm−αkww!xw−∑k,m,w=0∞−1k+m+w1β+k−1kαk+1β+m−1m×β−k+1/βλwm+αk+1/βww!e−λm+αk+1/βxw.


### 3.2. Quantile Function

The quantile function (QF) of the DEOWE distribution is the inverse function of the CDF, and it is given as follows:(16)xu=log1−p−β−1/β1/α+1λ−1.The three quarterlies (Q) of the DEOWE distribution can be obtained by settingu=0.25,0.5, and 0.75 in equation (11).Bowley’s skewness (BS) and Moor’s kurtosis (MK) can be calculated by the QF, respectively, as follows:(17)BS=Q1/4+Q3/4−2Q1/2Q3/4−Q1/4and(18)MK=Q7/8−Q5/8+Q3/8−Q1/8Q6/8−Q2/8.Table1 shows the numerical mean, variance, BS, and MK for the distribution using different parameters. These different values are coherent with the plots in Figure 1Table 1
Numerical values of mean, variance, BS, and MK of DEOWE distribution.
αβλMeanVarianceBSMK0.250.50.752.444813.689520.73314071.414710.50.751.50.83471.4065170.45183311.2972480.52.50.754.649141.728940.45308851.2651272.50.50.252.70521.2978230.041437681.2511051.5320.74230.92478320.34900831.44634921.50.51.79741.7205250.19503391.36669630.251.50.30230.2109358−0.011236121.2355230.75230.60830.86255740.400 86651.3505320.030.0532.645721.26830.999 99852.105148


## 3.1. Linear Representation

In this section, we made a linear representation for the PMF of the proposed distribution. We used linear representation to derive different statistical properties of the proposed model. Unfortunately, we reach a result form which does not follow any statistical model, and it is mathematically difficult to use to derive different statistical properties. In the case of the proposed distribution, we have three different cases for this linear representation.For|x|<1, we have the following expansion:(7)1+x−n=∑k=0∞−1kn+k−1kxk.For|x|>1, we have the following expansion:(8)1+x−n=∑k=0∞−1kn+k−1kx−k+n.Case 1.
Ifβeλx−1α<1 and x≠0, then we have(9)1+βeλx−1α−1/β=∑k,m,w=0∞−1k+m+w1/β+k−1kαkmβkλwm−αkww!xw,and if βeλx+1−1α<1 and x≠0, then we have(10)1+βeλx+1−1α−1/β=∑k,m,w=0∞−1k+m+w1/β+k−1kαkmβkλwm−αkww!e−λm−αkxw.
From equations (7) and (8), we have a linear representation of PMF (5) as the following:(11)PX=x;α,β,λ=∑k,m,w=0∞Φk,m,wxw,where Φk,m,w=−1k+m+w1/β+k−1kαkmβkλwm−αkw/w!1−e−λm−αk.Case 2.
. Ifβeλx−1α>1, then we have(12)1+βeλx−1α−1/β=∑k,m,w=0∞−1k+m+w1β+k−1kαk+1β+m−1m×β−k+1/βλwm+αk+1/βww!xw.
Ifβeλx+1−1α>1, then we have(13)1+βeλx+1−1α−1/β=∑k,m,w=0∞−1k+m+w1β+k−1kαk+1β+m−1m×β−k+1/βλwm+αk+1/βww!e−λm+αk+1/βxw.
From equations (9) and (10), we have a linear representation of PMF (5) as the following:(14)PX=x;α,β,λ=∑k,m,w=0∞Ψk,m,wxw,where Ψk,m,w=−1k+m+w1/β+k−1kαk+1/β+m−1mβ−k+1/βλwm+αk+1/βw/w!1−e−λm+αk+1/β.Case 3.
Ifβeλx−1α<1 and βeλx+1−1α>1, then, from equations (7) and (10), we have a linear representation of PMF (5) as the following:(15)PX=x;α,β,λ=∑k,m,w=0∞−1k+m+w1β+k−1kαkmβkλwm−αkww!xw−∑k,m,w=0∞−1k+m+w1β+k−1kαk+1β+m−1m×β−k+1/βλwm+αk+1/βww!e−λm+αk+1/βxw.


## 3.2. Quantile Function

The quantile function (QF) of the DEOWE distribution is the inverse function of the CDF, and it is given as follows:(16)xu=log1−p−β−1/β1/α+1λ−1.The three quarterlies (Q) of the DEOWE distribution can be obtained by settingu=0.25,0.5, and 0.75 in equation (11).Bowley’s skewness (BS) and Moor’s kurtosis (MK) can be calculated by the QF, respectively, as follows:(17)BS=Q1/4+Q3/4−2Q1/2Q3/4−Q1/4and(18)MK=Q7/8−Q5/8+Q3/8−Q1/8Q6/8−Q2/8.Table1 shows the numerical mean, variance, BS, and MK for the distribution using different parameters. These different values are coherent with the plots in Figure 1Table 1
Numerical values of mean, variance, BS, and MK of DEOWE distribution.
αβλMeanVarianceBSMK0.250.50.752.444813.689520.73314071.414710.50.751.50.83471.4065170.45183311.2972480.52.50.754.649141.728940.45308851.2651272.50.50.252.70521.2978230.041437681.2511051.5320.74230.92478320.34900831.44634921.50.51.79741.7205250.19503391.36669630.251.50.30230.2109358−0.011236121.2355230.75230.60830.86255740.400 86651.3505320.030.0532.645721.26830.999 99852.105148


## 4. Parameter Estimation

In this section, we use the MLE method to estimate the unknown parameters of the DEOWE distribution. Assume thatx1,…,xn represents a random count discrete sample that follows the DEOWE distribution having the parameters, α,β, and λ. So, the log-likelihood function will have the following form:(19)ℓΩ=∑i=1mln1+βeλx−1α−1/β−1+βeλx+1−1α−1/β,where Ω=α,β,λ is a vector of the DEOWE parameters. The MLEs are obtained by solving the following normal equations:(20)∂ℓΩ∂α=∑i=1nlneλ^xi+1−1eλ^xi+1−1α^1+β^eλ^xi+1−1α^−1/β^−11+β^eλ^xi−1α^−1/β^−1+β^eλ^xi+1−1α^−1/β^−lneλ^xi−1eλ^xi−1α^1+β^eλ^xi−1α^−1/β^−11+β^eλ^xi−1α^−1/β^−1+β^eλ^xi+1−1α^−1/β^=0,∂ℓΩ∂β=∑i=1nln1+β^eλ^xi−1α^1+β^eλ^xi−1α^−1/β^−eλ^xi−1α^1+β^eλ^xi−1α^−1/β^−11+β^eλ^xi−1α^−1/β^−1+β^eλ^xi+1−1α^−1/β^−ln1+β^eλ^xi+1−1α^1+β^eλ^xi+1−1α^−1/β^−eλ^xi+1−1α^1+β^eλ^xi+1−1α^−1/β^−11+β^eλ^xi−1α^−1/β^−1+β^eλ^xi+1−1α^−1/β^=0,and(21)∂ℓΩ∂λ=α^∑i=1nxi+1eλ^xi+11+β^eλ^xi+1−1α^−1/β^−1−xieλ^xi1+β^eλ^xi−1α^−1/β^−11+β^eλ^xi−1α^−1/β^−1+β^eλ^xi+1−1α^−1/β^=0.These equations cannot be solved explicitly. Hence, a nonlinear optimization algorithm as the Newton–Raphson method is used.


## 5. Simulation Studies

This part of the paper is devoted to make the Monte Carlo simulation procedure. This simulation study is performed for the classical estimation method: MLE for estimating parameters of DEOWE distribution in a lifetime by R language. Monte Carlo experiments are carried out based on data generated from 10 000 random samples from DEOWE distribution, whereX has DEOWE lifetime for different actual values of parameters and different sample sizes n: (20, 40, 70, and 100).We evaluate in every table Rbias and MSE of estimators.Tables2–4 summarize the simulation results of the point estimation method in this paper. We consider the Rbias and the MSE values to perform the needed comparison between different parameters’ values and their effect on point estimation values.Table 2
Rbias and MSE for parameter of DEOWE distribution by using MLE whenβ=0.5.
β=0.5λ0.010.050.15αNRbiasMSERbiasMSERbiasMSE0.520α0.049 40.006 330.165 80.025 40.305 30.077 3β−0.009 10.000 720.002 80.089 60.061 30.485 2λ0.057 80.000 0120.016 20.001 2−0.026 60.008 040α0.028 80.002 630.147 60.014 40.265 60.049 2β0.003 60.002 980.028 50.076 50.132 20.384 3λ0.034 30.000 005−0.051 30.000 2−0.047 10.006 370α0.037 00.002 010.141 70.010 20.250 70.029 9β−0.002 10.000 290.072 60.076 00.132 40.260 4λ0.004 50.000 002−0.041 30.0002−0.112 90.002 3100α0.040 00.001 9150.131 70.007 70.269 60.025 3β−0.003 00.000 0310.053 60.060 10.107 30.173 4λ−0.012 70.000 002−0.061 90.000 1−0.080 80.002 01.520α0.002 00.002 6740.048 20.045 50.155 60.150 5β−0.005 40.001 886−0.032 10.056 7−0.151 20.168 6λ0.010 60.000 002−0.022 20.000 053−0.100 80.000 840α−0.000 90.001 0180.010 70.004 80.136 30.084 6β0.001 00.000 389−0.007 90.004 5−0.022 40.077 4λ0.007 70.000 001−0.018 20.000 025−0.103 00.000 570α0.000 30.000 0520.042 50.016 50.150 80.080 0β−0.000 30.000 008−0.023 10.014 50.019 10.039 5λ0.004 30.000 001−0.031 40.000 018−0.102 80.000 4100α0.000 00.000 0020.024 10.007 00.102 20.037 7β0.000 00.000 0020.006 50.011 0−0.084 90.018 9λ−0.000 70.000 000 4−0.027 10.000 014−0.102 60.000 3520α−1.18E−061.63E−08−0.000 70.024 30.002 60.004 7β4.19E−051.64E−070.007 90.009 4−0.034 90.007 6λ−0.005 131.87E−07−0.031 116.91E−06−0.095 70.000 240α2.48E−091.41E−130.003 10.033 60.000 9497.01E−05β5.77E−081.21E−12−0.071 10.020 4−0.018 00.000 5λ−0.004 351.01E−07−0.036 066.71E−06−0.092 70.000 270α1.90E−055.52E−060.003 50.013 60.026 10.119 6β−0.000 66.23E−05−0.068 20.009 5−0.186 70.037 2λ−0.004 85.80E−08−0.035 984.63E−06−0.101 90.000 2100α−3.22E−098.23E−130.002 10.009 00.002 40.000 5β2.12E−073.87E−12−0.013 70.004 6−0.046 80.002 1λ−0.005 964.22E−08−0.032 873.62E−06−0.096 50.000 2Table 3
Rbias and MSE for parameter of DEOWE distribution by using MLE whenβ=1.5.
β=1.5λ0.050.150.5αnRbiasMSERbiasMSERbiasMSE0.520α0.150 70.026 70.255 40.060 20.410 20.120 0β−0.003 20.118 2−0.007 80.365 7−0.086 20.656 7λ0.045 10.003 4−0.021 30.012 9−0.212 80.080 840α0.143 90.016 10.239 10.035 10.428 80.101 8β−0.003 70.022 8−0.000 50.265 4−0.018 40.530 6λ−0.051 30.000 2−0.112 80.004 7−0.244 10.056 870α0.142 60.010 70.236 00.024 10.415 60.066 9β−0.001 90.022 90.004 20.108 7−0.016 20.215 7λ−0.082 50.000 1−0.149 50.001 4−0.271 60.032 2100α0.136 50.008 60.231 80.021 30.422 00.069 6β0.002 70.026 10.005 10.101 9−0.030 60.431 6λ−0.091 20.000 1−0.162 40.001 2−0.305 60.039 71.520α0.050 70.053 70.188 30.213 20.510 61.600 6β0.011 90.206 20.028 40.687 50.080 01.738 6λ−0.015 40.000 10−0.087 80.001 2−0.282 10.023 840α0.045 10.027 00.191 20.164 40.496 81.014 6β−0.002 30.051 60.050 50.326 10.079 71.076 8λ−0.030 10.000 05−0.090 90.000 7−0.276 70.022 270α0.024 90.007 30.173 50.104 10.495 80.835 9β−0.002 40.006 60.041 70.133 70.110 70.787 2λ−0.021 00.000 024−0.093 60.000 5−0.270 10.020 2100α0.023 30.006 60.131 90.051 40.442 30.567 4β0.000 80.014 10.009 50.057 60.066 90.397 7λ−0.028 60.000 021−0.101 40.000 4−0.275 50.020 6520α0.014 00.119 50.018 60.128 60.076 20.600 5β−0.048 20.079 0−0.022 90.085 8−0.198 80.461 0λ−0.040 085 61.08E−05−0.100 10.000 3−0.267 60.018 640α−0.000 60.024 20.013 10.046 10.052 70.128 7β−0.015 50.031 2−0.053 10.041 8−0.249 80.284 4λ−0.033 310 97.45E−06−0.099 50.000 3−0.269 00.018 570α0.008 40.009 90.002 50.000 70.051 20.124 7β−0.041 00.021 1−0.010 50.001 2−0.224 60.187 6λ−0.036 834 26.23E−06−0.096 80.000 2−0.269 70.018 4100α0.000 60.003 00.004 20.007 20.052 20.577 8β−0.003 40.006 2−0.014 50.005 4−0.160 60.173 5λ−0.034 227 14.87E−06−0.097 60.000 23−0.266 00.017 8Table 4
Rbias and MSE for parameter of DEOWE distribution by using MLE whenβ=5.
β=5λ0.150.51.5αnRbiasMSERbiasMSERbiasMSE0.520α0.252 20.083 10.418 60.189 20.950 40.588 5β0.005 11.021 80.017 11.796 80.080 24.372 6λ0.189 80.043 40.073 00.179 2−0.197 00.723 040α0.233 10.048 50.379 30.105 10.878 90.405 9β0.011 60.756 30.009 01.020 90.091 92.799 3λ−0.023 80.010 7−0.097 00.080 2−0.247 70.644 470α0.234 50.031 80.399 80.080 90.874 50.332 1β0.002 10.600 40.017 40.690 80.091 32.181 8λ−0.108 80.004 7−0.177 60.048 3−0.301 70.517 1100α0.233 70.026 00.392 90.067 60.883 60.297 9β0.004 70.238 60.013 00.609 80.118 21.928 7λ−0.141 40.002 1−0.202 30.037 8−0.342 30.463 81.520α0.242 50.274 50.521 31.015 60.833 82.409 0β0.042 01.460 50.072 53.340 7−0.013 71.945 9λ−0.090 80.001 6−0.259 60.031 7−0.427 00.669 740α0.183 00.137 50.484 20.760 10.852 92.258 9β0.011 10.280 10.072 02.463 6−0.035 71.429 4λ−0.102 30.001 0−0.278 30.025 4−0.470 20.662 970α0.179 40.125 30.468 90.701 40.847 41.938 5β0.013 60.275 80.047 42.127 6−0.028 11.145 3λ−0.103 90.000 7−0.274 20.021 4−0.508 40.599 1100α0.180 40.103 90.456 90.632 30.840 41.773 8β0.012 50.257 00.019 00.749 1−0.031 40.696 5λ−0.112 00.000 7−0.277 80.020 2−0.514 80.607 5520α0.040 10.277 50.126 00.679 00.272 72.233 7β−0.016 60.316 1−0.083 40.526 4−0.300 32.873 7λ−0.106 20.000 4−0.265 60.019 1−0.525 30.626 540α0.008 10.021 40.156 30.887 50.257 81.898 9β−0.004 90.014 4−0.091 30.628 0−0.272 02.189 7λ−0.100 40.000 3−0.271 80.019 1−0.525 10.623 170α0.060 80.334 80.116 80.532 60.246 91.819 5β−0.009 60.346 3−0.070 70.236 6−0.230 12.075 9λ−0.101 60.000 3−0.264 80.017 9−0.452 90.613 0100α0.031 60.114 00.106 30.581 50.238 81.514 2β−0.009 90.073 0−0.067 40.444 7−0.229 91.835 1λ−0.102 00.000 3−0.263 10.017 7−0.415 30.563 6In every table, we fix theβ value and increase the values of both λ and α, and then, we study the effect of increasing and decreasing the values. Concluding remarks are provided at the end of this section to illustrate the impact of the increment and decrements of the parameter’s values.


### 5.1. Concluding Remarks on Simulation Results

In this section of the paper, we introduce the major findings deduced from the simulation tables; we introduced the effect of increasing the sample sizes and the effect of increasing the true values of the parameters used in the simulation study. Also, we will discuss the effect of fixing the value of every two parameters and increasing the value of the third one. The following points can be noted from Tables2–4:(1)
As we can see from the results from Tables2–4, by increasing the sample size, we can see that the consistent property of MLEs comes true, and the Rbias value and MSE values of the three parameters decrease(2)
Referring to Table2, by making the value of β=0.5 and for a fixed value of α=0.5,1.5,5 and increasing λ from 0.01 to 0.15, we deduce that the MSE and Rbias of the parameters increase in most cases(3)
Referring to Table3 by fixing the value of β=1.5 and for a fixed value of α=0.5,1.5,5 and increasing λ from 0.05 to 0.5, we deduce that the MSE and Rbias of the parameters increase in most cases(4)
By increasing the value ofβ from 1.5 to be five as in Table 4 and making the sample size fixed for both values of beta, we deduced that the MSE and Rbias of the parameters increase in most cases


## 5.1. Concluding Remarks on Simulation Results

In this section of the paper, we introduce the major findings deduced from the simulation tables; we introduced the effect of increasing the sample sizes and the effect of increasing the true values of the parameters used in the simulation study. Also, we will discuss the effect of fixing the value of every two parameters and increasing the value of the third one. The following points can be noted from Tables2–4:(1)
As we can see from the results from Tables2–4, by increasing the sample size, we can see that the consistent property of MLEs comes true, and the Rbias value and MSE values of the three parameters decrease(2)
Referring to Table2, by making the value of β=0.5 and for a fixed value of α=0.5,1.5,5 and increasing λ from 0.01 to 0.15, we deduce that the MSE and Rbias of the parameters increase in most cases(3)
Referring to Table3 by fixing the value of β=1.5 and for a fixed value of α=0.5,1.5,5 and increasing λ from 0.05 to 0.5, we deduce that the MSE and Rbias of the parameters increase in most cases(4)
By increasing the value ofβ from 1.5 to be five as in Table 4 and making the sample size fixed for both values of beta, we deduced that the MSE and Rbias of the parameters increase in most cases


## 6. Applications to COVID-19 Data

In this section of the paper, we introduce two real data applications on the COVID-19 mortality numbers in Saudi Arabia, and the third data are outside Saudi Arabia; this third data were for Latvia mortality rate. The first data were an expressed sample on the first wave, while the second sample was an expressed sample on the second wave. The first application depends on the period from 26 December to 17 February 2021 for the infections in Saudi Arabia. We used this period because recording the infection numbers in this period was accurate as it was the peak of the second wave in Saudi Arabia. As in the earlier months of infection, recording the number of deaths was not accurate, so we choose this period specifically. The second dataset was taken for a period from 30 May 2020 to 20 August 2020. We choose this period because this period was the starting of the outbreak of COVID-19 in Saudi Arabia, and the mortality numbers start to increase also. This period is considered as the peak of the first wave in Saudi Arabia, which is very important to be modeled. We also evaluated the information criteria to introduce the importance of the proposed distribution compared with other competitors.


### 6.1. Application 1

In this section, we introduce a very important real data application for the DEOWE distribution, which the number of deaths due to COVID-19 infection in Saudi Arabia of 54 days of infection. Table5 contains some information and descriptive statistics for this data, which are recorded from 26 December to 17 February 2021. The data used in this application are as below: 9, 8, 9, 11, 8, 10, 9, 7, 9, 7, 10, 9, 7, 6, 4, 4, 5, 4, 5, 4, 6, 3, 5, 5, 6, 6, 3, 4, 4, 4, 2, 3, 4, 4, 3, 2, 4, 3, 4, 4, 3, 3, 4, 4, 5, 4, 4, 5, 5, 4, 4, 4, 6, 3. This data were collected from the world health organization, and these numbers represent the number of deaths per day. For more information, see the following link: https://covid19.who.int/. This data were used as a real data example for the proposed distribution and its competitor distributions. We compare the fitting results of the binomial (bionm), negative binomial(Nbionm), Poisson (Pois) distributions, see the work of Johnson et al. [22], discrete generalized exponential (DGE) distribution, see the work of Nekoukhou et al. [17], the discrete alpha power inverse Lomax (DAPIL) distribution is introduced by Almetwally and Ibrahim [23], the discrete Marshall–Olkin Generalized exponential (DMOGE) distribution is introduced by Almetwally et al. [19], and Skellam [24] introduced the Skellam distribution, and the results of these fitting are tabulated in Table 6.Table 5
Descriptive statistics for first data.
nMinQ1MedianMeanQ3MaxSkewnessKurtosis542.0004.0004.0005.2226.00011.0000.9302.855Table 6
MLE and standard error SE for models and goodness-of-fit criteria for each model.
Number of death/dayDeath countsBinomPoisDMOGEDAPLEDWDEOWENbinomSkellam000.369 40.291 30.000 10.003 50.000 00.000 20.299 10.291 4101.758 81.521 50.087 30.460 20.040 60.050 71.546 31.521 8224.265 13.972 82.169 73.362 72.226 91.395 14.004 83.973 4387.020 56.915 59.195 08.060 99.317 59.909 76.928 36.916 34198.821 99.028 613.532 010.883 512.813 215.329 19.006 89.029 2579.024 09.429 911.045 510.410 010.787 310.280 79.385 19.430 0657.824 98.207 57.085 37.987 87.342 96.049 78.165 08.207 2735.914 46.123 14.259 05.319 14.568 63.673 96.100 46.122 5823.976 83.997 02.551 33.240 42.745 92.324 33.995 83.996 4952.415 82.319 21.546 51.872 51.635 61.520 52.330 92.318 81021.342 11.211 20.949 31.052 20.976 91.021 01.226 01.210 91110.688 60.575 00.588 50.584 70.588 20.699 90.587 40.574 8α0.256 85.222 214.969 90.000 10.676 86.974 6517.031 55.222 0SE0.911 80.311 05.449 30.235 60.068 82.129 43.655 83.654 2β0.632 51.650 985.871 03.884 20.990 00.000 2SE0.091 90.000 21.673 40.155 50.009 9θ0.294 80.000 20.206 60.156 8SE0.318 40.037 00.011 0χ222.085 622.200 013.935 516.908 914.065 411.544 622.164 022.200 3P value0.023 70.022 90.236 60.110 60.229 40.398 80.023 10.022 9AIC236.283 2235.785 5230.480 2234.140 0230.824 3229.686 6237.816 7237.785 5CAIC236.360 1235.862 4230.960 2234.620 0231.304 3230.166 6238.052 0238.020 8BIC238.272 2237.774 5236.447 2240.106 9236.791 3235.653 6241.794 6241.763 5HQIC237.050 3236.552 6232.781 4236.441 2233.125 5231.987 9239.350 8239.319 7To make the comparison between many distributions, we must make this comparison based on some criteria; one of these analytical measures is called the Akaike information criterion (AIC), see [25]; there are another criteria called Bayesian information criterion (BIC), see [26], for more information, and we can also refer to Hannan–Quinn for more information criterion (HQIC), see [27], for more information, and last criteria are called the consistent Akaike information criterion (CAIC), see [28], for more details; all these criteria were used to compare the goodness of fit of the proposed model with other competing distributions. These measures are as follows.The AIC is given by(22)AIC=2k−2ℓ.The CAIC is(23)CAIC=2nkn−k−1−2ℓ.The BIC is calculated as follows:(24)BIC=klogn−2ℓ.The HQIC is(25)HQIC=2kloglogn−2ℓ.where k is the number of model parameters, n is the sample size, and ℓ refers to the log-likelihood function evaluated at the MLEs. Table 6 provides values of AIC, BIC, CAIC, HQIC and, chi square (χ2) with a degree of freedom, and its P value for all models is fitted based on the real dataset of Saudi Arabia. Figure 3 indicates a comparison between these distributions to get the best distribution; also, Figures 3–5 indicate the graphical plots of the data and the PMF of DEOWE distributions, with the corresponding competitive distributions with various numbers of parameters. As we can see that the plot in Figure 6 is the CDF of the distributions with the random variable X, while the third graph in Figure 7 is for the quantile function as a function of x, where x is the number of deaths per day; Figure 8 shows graphical plots of the data and the PMF of the DEOWE distributions.Figure 3
Graphical plots for the expected frequencies and the data using the PMF of different one parameter distributions, where thex-axis represents the number of deaths per day and the y-axis represents the frequency for this number.Figure 4
Graphical plots for the expected frequencies and the data using the PMF of different two-parameter distributions, where thex-axis represents the number of deaths per day and the y-axis represents the frequency for this number.Figure 5
Graphical plots for the expected frequencies and the data using the PMF of different three-parameter distributions, where thex-axis represents the number of deaths per day and the y-axis represents the frequency for this number.Figure 6
Graphical plots of the data and the CDF, as we can see in the plot, is distribution with the random variable with valuex, where x is the number of deaths per day.Figure 7
Graphical plots of the data and the quantile function as a function of (x), where x is the number of deaths per day.Figure 8
Graphical plots of the data and the PMF of the DEOWE distributions, wherex is the number of deaths per day and px is the probability for each x.


### 6.2. Application 2

In this section, the DEOWE distribution is fitted to another set of data of COVID-19 mortality numbers in Saudi Arabia of 83 days of infection, which is recorded from 30 May 2020 to 20 August 2020. Table7 contains some information and descriptive statistics for this data, while Table 8 contains the dataset used in this application associated with the frequency of each death number and the probability of this number; the data are as follows: 17, 22, 23, 22, 24, 30, 32, 31, 34, 36, 34, 37, 36, 38, 36, 39, 40, 39, 41, 39, 48, 45, 46, 37, 40, 39, 41, 41, 46, 37, 40, 48, 50, 49, 54, 50, 56, 58, 52, 49, 42, 41, 51, 30, 42, 20, 40, 42, 45, 37, 40, 39, 37, 34, 44, 34, 37, 31, 30, 27, 29, 27, 26, 24, 21, 30, 32, 35, 36, 35, 38, 37, 37, 32, 34, 36, 34, 35, 31, 39, 28, 34, 36. These data were collected from the World Health Organization and these numbers represents the number of deaths per day, for more information see the following link: https://covid19.who.int/. We compare the fitting results of the discrete generalized exponential (DGE) distribution, see the work of Nekoukhou et al. [17], the discrete Marshall–Olkin generalized exponential (DMOGE) distribution is introduced by Almetwally et al. [19], and exponentiated discrete Weibull (EDW) distribution is introduced by Nekoukhou et al. [29].Table 7
Descriptive statistics for the second data.
nMinQ1MedianMeanQ3MaxSkewnessKurtosis8317.0032.0037.0036.9341.0058.000.0933.029Table 8
The dataset used in this application associated with the frequency of each death number and the probability of this number.
Number of deaths/dayDeath counts for each numberDGEDMOGEEDWDEOWE1710.058 50.178 80.196 50.225 81800.122 00.225 40.273 30.293 71900.228 50.283 30.370 90.377 22010.389 40.355 20.492 00.478 82110.611 70.444 20.638 60.600 92220.895 60.553 90.812 20.746 02311.232 90.688 41.013 10.916 22421.608 70.852 01.240 21.113 42502.003 01.049 51.491 31.338 32612.393 81.285 01.762 51.590 92722.760 01.562 02.048 11.869 42813.083 51.882 12.341 42.170 02913.351 02.243 92.634 22.486 93043.554 22.641 82.917 52.811 43133.690 13.064 63.182 13.132 83233.759 83.494 63.418 63.437 93303.768 03.907 93.618 43.712 23473.721 44.275 83.774 33.940 93533.628 44.568 13.880 34.110 43663.497 94.757 83.932 94.209 73783.338 54.825 83.930 44.231 63823.158 34.765 13.873 54.174 13962.964 74.582 03.765 44.040 64052.763 84.294 93.610 93.839 54142.560 93.930 63.416 53.583 24232.360 33.519 13.189 93.286 64302.165 03.089 42.939 52.965 44411.977 62.665 72.673 82.634 94521.799 82.266 02.401 12.308 24621.632 51.901 92.128 91.996 14701.476 61.579 41.864 01.706 14821.332 11.300 01.611 81.443 24921.199 11.062 21.376 51.209 65021.077 10.862 81.161 21.005 85110.965 90.697 40.967 60.830 65210.864 80.561 50.796 50.681 85300.773 20.450 60.647 70.557 05410.690 50.360 80.520 30.453 15500.615 90.288 20.412 90.367 35610.548 90.229 90.323 70.297 05700.488 80.183 10.250 60.239 65810.434 90.145 70.191 60.193 0


### 6.3. Application 3

In this section, the DEOWE distribution is fitted to another set of data of COVID-19 mortality numbers in Latvia of 33 days of infection, which is recorded from 12 May 2021 to 13 April 2021. We choose this period specifically because it was the peak of the second wave of the COVID-19 infection in Latvia. Table9 contains some information and descriptive statistics for this data, while Table 10 contains the dataset used in this application associated with the frequency of each death number and the probability of this number, and Table 11 contains the MLE of the parameters and the P values and chi-square values for the distributions, also the information criteria for each distribution. The data are as follows: 11, 9, 11, 10, 2, 8, 12, 12, 10, 10, 5, 2, 12, 11, 13, 3, 5, 6, 5, 10, 6, 14, 9, 1, 8, 3, 3, 9, 17, 18, 5, 0, 4. These data were collected from the world health organization, and these numbers represent the number of deaths per day. For more information, see the following link: https://covid19.who.int/. We compare the fitting results of the discrete generalized exponential (DGE) distribution, see the work of Nekoukhou et al. [17], the discrete Marshall–Olkin generalized exponential (DMOGE) distribution is introduced by Almetwally et al. [19], and exponentiated discrete Weibull (EDW) distribution is introduced by Nekoukhou et al. [29].Table 9
Descriptive statistics for the data of COVID-19 mortality numbers in Latvia.
nMinQ1MedianMeanQ3MaxSkewnessKurtosis3305891118−0.509 940.189 237Table 10
The dataset used in this application associated with the frequency of each death number and the probability of this number.
ValueCountDGEDMOGEEDWDEOWE010.316 41.036 60.838 90.894 1111.316 71.112 91.416 11.427 3222.221 11.402 21.763 81.759 3332.785 11.750 82.029 32.018 3413.025 32.125 12.240 12.224 0543.023 42.481 12.403 12.380 2622.864 12.764 12.516 92.485 7702.615 72.921 22.576 82.537 4822.327 22.919 32.576 52.532 3932.031 02.758 52.510 92.468 01041.746 92.471 92.377 32.344 21131.486 02.111 82.178 32.163 61231.253 21.731 91.922 21.932 01311.049 71.373 21.624 31.659 81410.874 61.060 11.305 61.361 41500.725 60.801 50.990 71.055 01600.600 00.596 60.703 60.761 01710.494 80.438 90.463 30.500 31810.407 10.320 10.280 10.290 1Table 11
MLE and SE for models’ parameters and goodness-of-fit criteria for each model.
DGEDMOGEEDWDEOWEα2.752 70.672 13.625 21.338 7SE0.768 51.524 80.598 70.305 4β0.815 10.713 00.393 60.169 8SE0.029 8170.047 50.498 60.318 0θ23.465 10.999 90.065 9SE3.396 50.185 60.010 7χ215.249 311.719 011.700 811.700 5P value0.644 80.861 40.862 30.863 1AIC199.460 7196.707 7195.341 1195.325 5CACI199.860 7197.535 3196.168 7196.153 1BIC202.453 7201.197 2199.830 6199.815 0HQIC200.467 8198.218 3196.851 7196.836 1Figure 9
Graphical plots for the expected frequencies and the data using the PMF of different parameters distributions, where thex-axis represents the number of deaths per day and the y-axis represents the frequency for this number.Figure 10
Graphical plots of the data and the CDF, as we can see in the plot, the distributions with the random variable with valuex, where x is the number of deaths per day.Figure 11
Graphical plots of the data and the quantile function as a function of (x), where x is the number of deaths per day.Figure 12
Graphical plots of the data and the PMF of the DEOWE distributions, as we can see in the plot is the PMF of the distributions with the random variable with value (x), where x is the number of deaths per day and P(x) is the probability for x.Figure 13
Graphical plots for the expected frequencies and the data using the PMF of different parameters’ distributions, where thex-axis represents the number of deaths per day and the y-axis represents the frequency for this number.Figure 14
Graphical plots of the data and the CDF, as we can see in the plot, is the distributions with the random variable with valuex, where x is the number of deaths per day.Figure 15
Graphical plots of the data and the quantile function as a function ofx, where x is the number of deaths per day.Figure 16
Graphical plots of the data and the PMF of the DEOWE distributions, as we can see in the plot which is the PMF of the distributions with the random variable with valuex, where x is the number of deaths per day and P(x) is the probability for x.


### 6.4. Concluding Remarks on the Real Data

(1)
By referring to the goodness-of-fit measurements’ values in Tables6 and 12, we deduce that the DEOWE distribution has the lowest chi square, AIC, and CAIC values among all distributions for the three applications.(2)
By referring to the values of the goodness of fit measurements in Tables6 and 12, we deduce that the DEOWE distribution has the highest P value among all of its competitors for the three applications.(3)
For application one and by referring to Figures3 and 4, we can see that the one- and two-parameter distributions provide poor fitting for the data. In contrast, the three-parameter DEOWE distribution in Figure 5 provides better fitting for the data among all its competitors.(4)
For application two and by referring to Figure9, we can see that the three-parameter DEOWE distribution in Figure 9 provides better fitting for the data among all its competitors.(5)
For application two, we can see that the plot in Figure10 is the CDF of the distributions with the random variable X, while the graph in Figure 11 is for the quantile function as a function of x, where x is the number of deaths per day. Figure 12 shows graphical plots of the data and the PMF of the DEOWE distributions(6)
For application three and by referring to Figure13, we can see that the three-parameter DEOWE distribution in Figure 13 provides better fitting for the data among all its competitors for more information about the PMF of the other distributions, see the Appendix.(7)
For application three, we can see that the plot in Figure14 is the CDF of the distributions with the random variable X, while the graph in Figure 15 is for the quantile function as a function of x, where x is the number of deaths per day. Figure 16 shows graphical plots of the data and the PMF of the DEOWE distributions


## 6.1. Application 1

In this section, we introduce a very important real data application for the DEOWE distribution, which the number of deaths due to COVID-19 infection in Saudi Arabia of 54 days of infection. Table5 contains some information and descriptive statistics for this data, which are recorded from 26 December to 17 February 2021. The data used in this application are as below: 9, 8, 9, 11, 8, 10, 9, 7, 9, 7, 10, 9, 7, 6, 4, 4, 5, 4, 5, 4, 6, 3, 5, 5, 6, 6, 3, 4, 4, 4, 2, 3, 4, 4, 3, 2, 4, 3, 4, 4, 3, 3, 4, 4, 5, 4, 4, 5, 5, 4, 4, 4, 6, 3. This data were collected from the world health organization, and these numbers represent the number of deaths per day. For more information, see the following link: https://covid19.who.int/. This data were used as a real data example for the proposed distribution and its competitor distributions. We compare the fitting results of the binomial (bionm), negative binomial(Nbionm), Poisson (Pois) distributions, see the work of Johnson et al. [22], discrete generalized exponential (DGE) distribution, see the work of Nekoukhou et al. [17], the discrete alpha power inverse Lomax (DAPIL) distribution is introduced by Almetwally and Ibrahim [23], the discrete Marshall–Olkin Generalized exponential (DMOGE) distribution is introduced by Almetwally et al. [19], and Skellam [24] introduced the Skellam distribution, and the results of these fitting are tabulated in Table 6.Table 5
Descriptive statistics for first data.
nMinQ1MedianMeanQ3MaxSkewnessKurtosis542.0004.0004.0005.2226.00011.0000.9302.855Table 6
MLE and standard error SE for models and goodness-of-fit criteria for each model.
Number of death/dayDeath countsBinomPoisDMOGEDAPLEDWDEOWENbinomSkellam000.369 40.291 30.000 10.003 50.000 00.000 20.299 10.291 4101.758 81.521 50.087 30.460 20.040 60.050 71.546 31.521 8224.265 13.972 82.169 73.362 72.226 91.395 14.004 83.973 4387.020 56.915 59.195 08.060 99.317 59.909 76.928 36.916 34198.821 99.028 613.532 010.883 512.813 215.329 19.006 89.029 2579.024 09.429 911.045 510.410 010.787 310.280 79.385 19.430 0657.824 98.207 57.085 37.987 87.342 96.049 78.165 08.207 2735.914 46.123 14.259 05.319 14.568 63.673 96.100 46.122 5823.976 83.997 02.551 33.240 42.745 92.324 33.995 83.996 4952.415 82.319 21.546 51.872 51.635 61.520 52.330 92.318 81021.342 11.211 20.949 31.052 20.976 91.021 01.226 01.210 91110.688 60.575 00.588 50.584 70.588 20.699 90.587 40.574 8α0.256 85.222 214.969 90.000 10.676 86.974 6517.031 55.222 0SE0.911 80.311 05.449 30.235 60.068 82.129 43.655 83.654 2β0.632 51.650 985.871 03.884 20.990 00.000 2SE0.091 90.000 21.673 40.155 50.009 9θ0.294 80.000 20.206 60.156 8SE0.318 40.037 00.011 0χ222.085 622.200 013.935 516.908 914.065 411.544 622.164 022.200 3P value0.023 70.022 90.236 60.110 60.229 40.398 80.023 10.022 9AIC236.283 2235.785 5230.480 2234.140 0230.824 3229.686 6237.816 7237.785 5CAIC236.360 1235.862 4230.960 2234.620 0231.304 3230.166 6238.052 0238.020 8BIC238.272 2237.774 5236.447 2240.106 9236.791 3235.653 6241.794 6241.763 5HQIC237.050 3236.552 6232.781 4236.441 2233.125 5231.987 9239.350 8239.319 7To make the comparison between many distributions, we must make this comparison based on some criteria; one of these analytical measures is called the Akaike information criterion (AIC), see [25]; there are another criteria called Bayesian information criterion (BIC), see [26], for more information, and we can also refer to Hannan–Quinn for more information criterion (HQIC), see [27], for more information, and last criteria are called the consistent Akaike information criterion (CAIC), see [28], for more details; all these criteria were used to compare the goodness of fit of the proposed model with other competing distributions. These measures are as follows.The AIC is given by(22)AIC=2k−2ℓ.The CAIC is(23)CAIC=2nkn−k−1−2ℓ.The BIC is calculated as follows:(24)BIC=klogn−2ℓ.The HQIC is(25)HQIC=2kloglogn−2ℓ.where k is the number of model parameters, n is the sample size, and ℓ refers to the log-likelihood function evaluated at the MLEs. Table 6 provides values of AIC, BIC, CAIC, HQIC and, chi square (χ2) with a degree of freedom, and its P value for all models is fitted based on the real dataset of Saudi Arabia. Figure 3 indicates a comparison between these distributions to get the best distribution; also, Figures 3–5 indicate the graphical plots of the data and the PMF of DEOWE distributions, with the corresponding competitive distributions with various numbers of parameters. As we can see that the plot in Figure 6 is the CDF of the distributions with the random variable X, while the third graph in Figure 7 is for the quantile function as a function of x, where x is the number of deaths per day; Figure 8 shows graphical plots of the data and the PMF of the DEOWE distributions.Figure 3
Graphical plots for the expected frequencies and the data using the PMF of different one parameter distributions, where thex-axis represents the number of deaths per day and the y-axis represents the frequency for this number.Figure 4
Graphical plots for the expected frequencies and the data using the PMF of different two-parameter distributions, where thex-axis represents the number of deaths per day and the y-axis represents the frequency for this number.Figure 5
Graphical plots for the expected frequencies and the data using the PMF of different three-parameter distributions, where thex-axis represents the number of deaths per day and the y-axis represents the frequency for this number.Figure 6
Graphical plots of the data and the CDF, as we can see in the plot, is distribution with the random variable with valuex, where x is the number of deaths per day.Figure 7
Graphical plots of the data and the quantile function as a function of (x), where x is the number of deaths per day.Figure 8
Graphical plots of the data and the PMF of the DEOWE distributions, wherex is the number of deaths per day and px is the probability for each x.


## 6.2. Application 2

In this section, the DEOWE distribution is fitted to another set of data of COVID-19 mortality numbers in Saudi Arabia of 83 days of infection, which is recorded from 30 May 2020 to 20 August 2020. Table7 contains some information and descriptive statistics for this data, while Table 8 contains the dataset used in this application associated with the frequency of each death number and the probability of this number; the data are as follows: 17, 22, 23, 22, 24, 30, 32, 31, 34, 36, 34, 37, 36, 38, 36, 39, 40, 39, 41, 39, 48, 45, 46, 37, 40, 39, 41, 41, 46, 37, 40, 48, 50, 49, 54, 50, 56, 58, 52, 49, 42, 41, 51, 30, 42, 20, 40, 42, 45, 37, 40, 39, 37, 34, 44, 34, 37, 31, 30, 27, 29, 27, 26, 24, 21, 30, 32, 35, 36, 35, 38, 37, 37, 32, 34, 36, 34, 35, 31, 39, 28, 34, 36. These data were collected from the World Health Organization and these numbers represents the number of deaths per day, for more information see the following link: https://covid19.who.int/. We compare the fitting results of the discrete generalized exponential (DGE) distribution, see the work of Nekoukhou et al. [17], the discrete Marshall–Olkin generalized exponential (DMOGE) distribution is introduced by Almetwally et al. [19], and exponentiated discrete Weibull (EDW) distribution is introduced by Nekoukhou et al. [29].Table 7
Descriptive statistics for the second data.
nMinQ1MedianMeanQ3MaxSkewnessKurtosis8317.0032.0037.0036.9341.0058.000.0933.029Table 8
The dataset used in this application associated with the frequency of each death number and the probability of this number.
Number of deaths/dayDeath counts for each numberDGEDMOGEEDWDEOWE1710.058 50.178 80.196 50.225 81800.122 00.225 40.273 30.293 71900.228 50.283 30.370 90.377 22010.389 40.355 20.492 00.478 82110.611 70.444 20.638 60.600 92220.895 60.553 90.812 20.746 02311.232 90.688 41.013 10.916 22421.608 70.852 01.240 21.113 42502.003 01.049 51.491 31.338 32612.393 81.285 01.762 51.590 92722.760 01.562 02.048 11.869 42813.083 51.882 12.341 42.170 02913.351 02.243 92.634 22.486 93043.554 22.641 82.917 52.811 43133.690 13.064 63.182 13.132 83233.759 83.494 63.418 63.437 93303.768 03.907 93.618 43.712 23473.721 44.275 83.774 33.940 93533.628 44.568 13.880 34.110 43663.497 94.757 83.932 94.209 73783.338 54.825 83.930 44.231 63823.158 34.765 13.873 54.174 13962.964 74.582 03.765 44.040 64052.763 84.294 93.610 93.839 54142.560 93.930 63.416 53.583 24232.360 33.519 13.189 93.286 64302.165 03.089 42.939 52.965 44411.977 62.665 72.673 82.634 94521.799 82.266 02.401 12.308 24621.632 51.901 92.128 91.996 14701.476 61.579 41.864 01.706 14821.332 11.300 01.611 81.443 24921.199 11.062 21.376 51.209 65021.077 10.862 81.161 21.005 85110.965 90.697 40.967 60.830 65210.864 80.561 50.796 50.681 85300.773 20.450 60.647 70.557 05410.690 50.360 80.520 30.453 15500.615 90.288 20.412 90.367 35610.548 90.229 90.323 70.297 05700.488 80.183 10.250 60.239 65810.434 90.145 70.191 60.193 0


## 6.3. Application 3

In this section, the DEOWE distribution is fitted to another set of data of COVID-19 mortality numbers in Latvia of 33 days of infection, which is recorded from 12 May 2021 to 13 April 2021. We choose this period specifically because it was the peak of the second wave of the COVID-19 infection in Latvia. Table9 contains some information and descriptive statistics for this data, while Table 10 contains the dataset used in this application associated with the frequency of each death number and the probability of this number, and Table 11 contains the MLE of the parameters and the P values and chi-square values for the distributions, also the information criteria for each distribution. The data are as follows: 11, 9, 11, 10, 2, 8, 12, 12, 10, 10, 5, 2, 12, 11, 13, 3, 5, 6, 5, 10, 6, 14, 9, 1, 8, 3, 3, 9, 17, 18, 5, 0, 4. These data were collected from the world health organization, and these numbers represent the number of deaths per day. For more information, see the following link: https://covid19.who.int/. We compare the fitting results of the discrete generalized exponential (DGE) distribution, see the work of Nekoukhou et al. [17], the discrete Marshall–Olkin generalized exponential (DMOGE) distribution is introduced by Almetwally et al. [19], and exponentiated discrete Weibull (EDW) distribution is introduced by Nekoukhou et al. [29].Table 9
Descriptive statistics for the data of COVID-19 mortality numbers in Latvia.
nMinQ1MedianMeanQ3MaxSkewnessKurtosis3305891118−0.509 940.189 237Table 10
The dataset used in this application associated with the frequency of each death number and the probability of this number.
ValueCountDGEDMOGEEDWDEOWE010.316 41.036 60.838 90.894 1111.316 71.112 91.416 11.427 3222.221 11.402 21.763 81.759 3332.785 11.750 82.029 32.018 3413.025 32.125 12.240 12.224 0543.023 42.481 12.403 12.380 2622.864 12.764 12.516 92.485 7702.615 72.921 22.576 82.537 4822.327 22.919 32.576 52.532 3932.031 02.758 52.510 92.468 01041.746 92.471 92.377 32.344 21131.486 02.111 82.178 32.163 61231.253 21.731 91.922 21.932 01311.049 71.373 21.624 31.659 81410.874 61.060 11.305 61.361 41500.725 60.801 50.990 71.055 01600.600 00.596 60.703 60.761 01710.494 80.438 90.463 30.500 31810.407 10.320 10.280 10.290 1Table 11
MLE and SE for models’ parameters and goodness-of-fit criteria for each model.
DGEDMOGEEDWDEOWEα2.752 70.672 13.625 21.338 7SE0.768 51.524 80.598 70.305 4β0.815 10.713 00.393 60.169 8SE0.029 8170.047 50.498 60.318 0θ23.465 10.999 90.065 9SE3.396 50.185 60.010 7χ215.249 311.719 011.700 811.700 5P value0.644 80.861 40.862 30.863 1AIC199.460 7196.707 7195.341 1195.325 5CACI199.860 7197.535 3196.168 7196.153 1BIC202.453 7201.197 2199.830 6199.815 0HQIC200.467 8198.218 3196.851 7196.836 1Figure 9
Graphical plots for the expected frequencies and the data using the PMF of different parameters distributions, where thex-axis represents the number of deaths per day and the y-axis represents the frequency for this number.Figure 10
Graphical plots of the data and the CDF, as we can see in the plot, the distributions with the random variable with valuex, where x is the number of deaths per day.Figure 11
Graphical plots of the data and the quantile function as a function of (x), where x is the number of deaths per day.Figure 12
Graphical plots of the data and the PMF of the DEOWE distributions, as we can see in the plot is the PMF of the distributions with the random variable with value (x), where x is the number of deaths per day and P(x) is the probability for x.Figure 13
Graphical plots for the expected frequencies and the data using the PMF of different parameters’ distributions, where thex-axis represents the number of deaths per day and the y-axis represents the frequency for this number.Figure 14
Graphical plots of the data and the CDF, as we can see in the plot, is the distributions with the random variable with valuex, where x is the number of deaths per day.Figure 15
Graphical plots of the data and the quantile function as a function ofx, where x is the number of deaths per day.Figure 16
Graphical plots of the data and the PMF of the DEOWE distributions, as we can see in the plot which is the PMF of the distributions with the random variable with valuex, where x is the number of deaths per day and P(x) is the probability for x.


## 6.4. Concluding Remarks on the Real Data

(1)
By referring to the goodness-of-fit measurements’ values in Tables6 and 12, we deduce that the DEOWE distribution has the lowest chi square, AIC, and CAIC values among all distributions for the three applications.(2)
By referring to the values of the goodness of fit measurements in Tables6 and 12, we deduce that the DEOWE distribution has the highest P value among all of its competitors for the three applications.(3)
For application one and by referring to Figures3 and 4, we can see that the one- and two-parameter distributions provide poor fitting for the data. In contrast, the three-parameter DEOWE distribution in Figure 5 provides better fitting for the data among all its competitors.(4)
For application two and by referring to Figure9, we can see that the three-parameter DEOWE distribution in Figure 9 provides better fitting for the data among all its competitors.(5)
For application two, we can see that the plot in Figure10 is the CDF of the distributions with the random variable X, while the graph in Figure 11 is for the quantile function as a function of x, where x is the number of deaths per day. Figure 12 shows graphical plots of the data and the PMF of the DEOWE distributions(6)
For application three and by referring to Figure13, we can see that the three-parameter DEOWE distribution in Figure 13 provides better fitting for the data among all its competitors for more information about the PMF of the other distributions, see the Appendix.(7)
For application three, we can see that the plot in Figure14 is the CDF of the distributions with the random variable X, while the graph in Figure 15 is for the quantile function as a function of x, where x is the number of deaths per day. Figure 16 shows graphical plots of the data and the PMF of the DEOWE distributions


## 7. Conclusion

In this paper, we introduced a new distribution, which is called DEOWE distribution the aim to do this work was the lack of flexibility in other distributions. We studied its statistical properties and obtained a linear representation for its PMF and the associated quantile function. We used the MLE method for estimating the distribution parametersα,β, and λ. Also, a real dataset of the mortality numbers in the Kingdom of Saudi Arabia (KSA) was considered to assess the performance of the DEOWE. The distribution fitting for the real dataset was compared with its competitors, and by referring to the values of the goodness of fit measurements, we deduce that the DEOWE distribution has the lowest chi square, AIC, and CAIC for the first dataset, and for the second dataset, we deduce that the DEOWE distribution has the lowest chi square, AIC, CAIC, BIC, and HQIC and the highest P value among all of its competitors. This result indicates that the DEOWE distribution provides a superior model for fitting the mortality number compared with other competitive distributions. Also, we make a graphical plot for the data using the DEOWE with other competitive distributions, and the plots come in our favor and assure the results of the goodness-of-fit measurements.Table 12
MLE and SE for models’ parameters and goodness-of-fit criteria for each model.
DGEDMOGEEDWDEOWEα0.884 720.505 03.157 64.835 5SE0.008 90.003 91.516 50.767 1β57.948 80.792 62.440 60.570 4SE18.469 40.004 00.951 50.315 7θ300.010 00.999 80.018 1SE0.002 70.000 50.000 7χ252.624 243.500 338.782 637.463 0P value0.105 40.365 40.569 60.628 6AIC602.986 2594.508 2593.607 3593.432 5CAIC603.136 2594.812 0593.911 1593.736 3BIC607.823 8601.764 7600.863 9600.689 1HQIC604.929 7597.423 4596.522 6596.347 8



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*Source: 7192833-2021-12-24.xml*