A Novel Discrete Linear-Exponential Distribution for Modeling Physical and Medical Data

Abstract

In real-life data, count data are considered more significant in different fields. In this article, a new form of the one-parameter discrete linear-exponential distribution is derived based on the survival function as a discretization technique. An extensive study of this distribution is conducted under its new form, including characteristic functions and statistical properties. It is shown that this distribution is appropriate for modeling over-dispersed count data. Moreover, its probability mass function is right-skewed with different shapes. The unknown model parameter is estimated using the maximum likelihood method, with more attention given to Bayesian estimation methods. The Bayesian estimator is computed based on three different loss functions: a square error loss function, a linear exponential loss function, and a generalized entropy loss function. The simulation study is implemented to examine the distribution’s behavior and compare the classical and Bayesian estimation methods, which indicated that the Bayesian method under the generalized entropy loss function with positive weight is the best for all sample sizes with the minimum mean squared errors. Finally, the discrete linear-exponential distribution proves its efficiency in fitting discrete physical and medical lifetime count data in real-life against other related distributions.

1. Introduction

Continuous and discrete lifetime distributions are important and extensively used in analyzing and modeling data in various applied sciences, such as economics, engineering, finance, medical and biological sciences, etc. Despite the fact that most real-life data are continuous, they are discrete in observation. Consequently, it is important to obtain new discrete distributions to fit different types of lifetime discrete data. One of the most important recent methods to create discrete distributions is discretization techniques. Although several discretization techniques have been adopted extensively in the literature, the survival function (SF) is considered one of the most common discretization techniques used. Recently, some important discrete lifetime models were introduced by [1,2,3,4,5,6,7,8,9,10].

As is known, the classical distributions with one parameter, such as exponential, Lindley, and other distributions, are popular in modeling lifetime data. These models draw researchers’ attention because of their several desirable features and physical interpretations. Therefore, the extension of these distributions has become a field for a wide range of statistical literature such as [11,12]. However, a comparative study by [13] shows that both exponential and Lindley distributions are not suitable for many lifetime data due to their shapes, hazard rate functions, and mean residual life functions, among others, and as a result, the need has arisen to develop more flexible distributions that fit different discrete survival datasets.

A One-Parameter Linear-Exponential distribution (OPLE) was recently introduced by [14], based on the product of a linear ($x+{\varphi }^2$) and exponential function $e^{-\varphi x}$ with a single parameter $\varphi >0$. According to [14], the OPLE distribution is a better alternative to the Lindley (1958) distribution, as it is more suitable for data sets with variance greater than the mean. The importance of this distribution motivates us to review this distribution and generate its discrete analog. The probability density function (PDF) of the OPLE distribution could be expressed as:

$$f(x,\varphi )={\displaystyle \frac{{\varphi }^2}{1+{\varphi }^3}}({\varphi }^2+x)e^{-\varphi x};x>0,\varphi >0.$$

(1)

The corresponding SF has the form:

$$S(x,\varphi )={\displaystyle \frac{(1+\varphi x+{\varphi }^3)}{1+{\varphi }^3}}{e}^{-\varphi x};x>0,\varphi >0.$$

(2)

Ref. [8] proposed a discrete analog of the OPLE distribution, and its properties were calculated. He estimated parameters using different classical estimation approaches. Furthermore, four real data sets were analyzed to demonstrate the importance of the proposed distribution.

In this study, a one-parameter discrete linear-exponential distribution (DLE) is introduced in a new model and compared to [8], as well as its probability mass function (PMF) and cumulative distribution function (CDF). A comprehensive study of DLE distribution is conducted, including obtaining some important characteristic functions and statistical properties. The maximum likelihood (ML) and Bayesian estimation methods are used to estimate the unknown model parameter. In particular, the Bayesian estimation method is extensively addressed using three different loss functions: square error loss function (SELF), linear-exponential loss function (LINEX), and general entropy loss function (GELF) based on two cases: gamma informative priors and uniform non-informative prior. Using Markov Chain Monte Carlo (MCMC), the Metropolis–Hasting algorithm is implemented to generate samples from the posterior distributions and compute the Bayesian estimators.

The rest of the paper is organized as follows: In Section 2, we introduce the DLE distribution and derive different mathematical properties. In Section 3, ML and Bayesian estimation methods are derived. The simulation study results and real data analysis are illustrated in Section 4 and Section 5, respectively. Finally, in Section 6, we provide some conclusions.

2. Some Properties of DLE Distribution

The continuous random variable (CRV) is identified either by its PDF, moment generating function (MGF), moments, hazard rate function (HRF), etc. Primarily, deriving a discrete analog from a continuous distribution is based on the principle of keeping one or more characteristic properties of the continuous one. There are different techniques in which a discrete analog of a discrete random variable (DRV) X of a CRV Y can be generated according to [13], such as discretization methods based on SF, PDF, CDF, HRF, reversed-HRF, the difference equation analog of a Persian differential equation and a two-stage composite method. This study aims to use an SF technique since it is the most common in the literature ([2,3,4,6,7], etc.). Let Y be a CRV with SF $S(y,\varphi )$, then the PMF of the DRV $X=\left[y\right]$, which is the largest integer less than or equal to Y derived using the SF technique and is written as follows:

$$P(X=x)=S(x,\varphi )-S(x+1,\varphi );x=0,1,2,\dots .$$

(3)

The feature of this method is that the SF for both continuous and discrete distributions are identical. If the random variable Y follows OPLE distribution then the PMF of the DLE distribution can be written as:

$$p(X,\varphi )={\displaystyle \frac{e^{-\varphi x}(1+\varphi x+{\varphi }^3)-e^{-\varphi (x+1)}(1+\varphi (x+1)+{\varphi }^3)}{1+{\varphi }^3}};x=0,1,2\dots ,\varphi >0.$$

(4)

The corresponding CDF is given by:

$$F(X,\varphi )=\sum _{x=0}^{x}f(X,\varphi ),$$

$$F(X,\varphi )=1-{\displaystyle \frac{e^{-\varphi (x+1)}}{1+{\varphi }^3}}(1+\varphi (x+1)+{\varphi }^3)x=0,1,2,\dots .\varphi >0.$$

(5)

Figure 1 shows that the PMF of the DLE distribution can be right skewed depending on the selected values of the parameters. The mode of the DLE distribution moves towards the left for large values of $\varphi$.

2.1. Survival and Hazard Rate Functions

The SF of the DLE distribution is given by:

$$S(X,\varphi )=P(X⩾x,\varphi )=1-F(X,\varphi )+P(X=x,\varphi );X=0,1,2\dots .$$

$$S(X,\varphi )={\displaystyle \frac{(1+\varphi x+{\varphi }^3)}{1+{\varphi }^3}}{e}^{-\varphi x};x=0,1,2,\dots ,\varphi >0.$$

(6)

The corresponding HRF is given by:

$$h(X,\varphi )={\displaystyle \frac{f(X,\varphi )}{S(X,\varphi )}}.$$

$$h(X,\varphi )=1-{\displaystyle \frac{e^{-\varphi }(1+\varphi (x+1)+{\varphi }^3)}{(1+\varphi x+{\varphi }^3)}};x=0,1,2,\dots ,\varphi >0.$$

(7)

Figure 2 illustrates that the behavior of the HRF of the DLE distribution is increasing for different values of $\varphi$.

2.2. Moment Generating Function (MGF)

The MGF of DLE distribution can be written as:

$$M_X\left(t\right)=\sum _{x=0}^{\infty }{e}^{tx}P(x,\varphi ).$$

$$M_X\left(t\right)=1+(e^t-1)\sum _{x=1}^{\infty }{e}^{t(X-1)}{\displaystyle \frac{(1+\varphi x+{\varphi }^3)}{1+{\varphi }^3}}{e}^{-\varphi x};x=0,1,2,\dots ,\varphi >0.$$

(8)

2.3. Probability Generating Function (PGF)

The PGF as well as its rth moment are investigated in this section. The moments of a probability distribution are important for measuring its different properties such as mean, variance, skewness, kurtosis, etc. Assume the random variable X has a DRV of DLE distribution, then the PGF can be expressed as:

$$G_x\left(z\right)=\sum _{x=0}^{\infty }{z}^{x}P(x,\varphi )$$

$$\begin{array}{cc}\hfill G_x\left(z\right)& =\sum _{x=0}^{\infty }{z}^x{\displaystyle \frac{e^{-\varphi x}(1+\varphi x+{\varphi }^3)-e^{-\varphi (x+1)}(1+\varphi (x+1)+{\varphi }^3)}{1+{\varphi }^3}}\hfill \\ \hfill G_x\left(z\right)& ={\displaystyle \frac{(1+{\varphi }^3)-e^{-\varphi }(1+\varphi +{\varphi }^3)}{(1+{\varphi }^3)}}+z{\displaystyle \frac{(1+\varphi +{\varphi }^3)-e^{-2\varphi }(1+2\varphi +{\varphi }^3)}{(1+{\varphi }^3)}}+z^2{\displaystyle \frac{(1+2\varphi +{\varphi }^3)-e^{-3\varphi }(1+3\varphi +{\varphi }^3)}{(1+{\varphi }^3)}}\dots .\hfill \end{array}$$

$$G_x\left(z\right)=1+(z-1)\sum _{x=1}^{\infty }{z}^{x-1}{\displaystyle \frac{(1+\varphi x+{\varphi }^3)}{1+{\varphi }^3}}{e}^{-\varphi x}$$

(9)

The first four factorial moments of the DLE distribution can be obtained by differentiating the $G_x\left(z\right)$ with respect to z and setting z = 1, as follows:

$$\begin{array}{cc}\hfill G_x^{\prime }{\left(z\right)|}_{z=1}& =\sum _{x=1}^{\infty }{\displaystyle \frac{(1+\varphi x+{\varphi }^3)}{1+{\varphi }^3}}{e}^{-\varphi x}.\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill G_x^{\prime \prime }{\left(z\right)|}_{z=1}& =2\sum _{x=1}^{\infty }(x-1){\displaystyle \frac{(1+\varphi x+{\varphi }^3)}{1+{\varphi }^3}}{e}^{-\varphi x}.\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill G_x^{\prime \prime \prime }{\left(z\right)|}_{z=1}& =3\sum _{x=1}^{\infty }(x-1)(x-2){\displaystyle \frac{(1+\varphi x+{\varphi }^3)}{1+{\varphi }^3}}{e}^{-\varphi x}.\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill G_x^{\prime \prime \prime \prime }{\left(z\right)|}_{z=1}& =4\sum _{x=1}^{\infty }(x-1)(x-2)(x-3){\displaystyle \frac{(1+\varphi x+{\varphi }^3)}{1+{\varphi }^3}}{e}^{-\varphi x}.\hfill \end{array}$$

(13)

Regarding the original moments, the first four moments of the DLE distribution can be calculated using the factorial moments in Equations (10)–(13) by substituting r = 1, 2, 3, 4 in Equation (14).

$$E\left[{\left(x\right)}_r\right]=E[x(x-1)(x-2)\dots \dots .(x-r+1)].$$

(14)

Thus, the first four original moments are revealed in Equations (15)–(18), respectively.

$$\begin{array}{cc}\hfill {\mu }_1^{\prime }& =\sum _{x=1}^{\infty }{\displaystyle \frac{(1+\varphi x+{\varphi }^3)}{1+{\varphi }^3}}{e}^{-\varphi x}.\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill {\mu }_2^{\prime }& =\sum _{x=1}^{\infty }(2x-1){\displaystyle \frac{(1+\varphi x+{\varphi }^3)}{1+{\varphi }^3}}{e}^{-\varphi x}.\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill {\mu }_3^{\prime }& =\sum _{x=1}^{\infty }(3x^2-3x+1){\displaystyle \frac{(1+\varphi x+{\varphi }^3)}{1+{\varphi }^3}}{e}^{-\varphi x}.\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill {\mu }_4^{\prime }& =\sum _{x=1}^{\infty }(4x^3-6x^2+4x-1){\displaystyle \frac{(1+\varphi x+{\varphi }^3)}{1+{\varphi }^3}}{e}^{-\varphi x}.\hfill \end{array}$$

(18)

Unfortunately, the first four original moments are not in closed form, as a result, the Wolfram-cloud program is used to handle this issue. The closed form is given as follows:

$$\begin{array}{cc}\hfill {\mu }_1^{\prime }=& {\displaystyle \frac{-1+e^{\varphi }+\varphi e^{\varphi }-{\varphi }^3+{\varphi }^3{e}^{\varphi }}{{(-1+e^{\varphi })}^2(1+{\varphi }^3)}}.\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill {\mu }_2^{\prime }=& {\displaystyle \frac{-1+e^{2\varphi }+3\varphi e^{\varphi }+\varphi e^{2\varphi }-{\varphi }^3+{\varphi }^3{e}^{2\varphi }}{{(-1+e^{\varphi })}^3(1+{\varphi }^3)}}.\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill {\mu }_3^{\prime }=& {\displaystyle \frac{-1-3e^{\varphi }+3e^{2\varphi }+e^{3\varphi }+7\varphi e^{\varphi }+10\varphi e^{2\varphi }+\varphi e^{3\varphi }}{{(-1+e^{\varphi })}^4(1+{\varphi }^3)}}\hfill \\ & +{\displaystyle \frac{{\varphi }^3-3{\varphi }^3{e}^{\varphi }+3{\varphi }^3{e}^{2\varphi }+{\varphi }^3{e}^{3\varphi }}{{(-1+e^{\varphi })}^4(1+{\varphi }^3)}}.\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill {\mu }_4^{\prime }=& {\displaystyle \frac{-1-10e^{\varphi }+10e^{3\varphi }+e^{4\varphi }+15\varphi e^{\varphi }+55\varphi e^{2\varphi }+25\varphi e^{3\varphi }+\varphi e^{4\varphi }}{{(-1+e^{\varphi })}^5(1+{\varphi }^3)}}\hfill \\ & +{\displaystyle \frac{-{\varphi }^3-10{\varphi }^3{e}^{\varphi }+10{\varphi }^3{e}^{3\varphi }+{\varphi }^3{e}^{4\varphi }}{{(-1+e^{\varphi })}^5(1+{\varphi }^3)}}.\hfill \end{array}$$

(22)

From Equations (19)–(22), the first four moments about the mean can be simply derived as:

$${\mu }_1=0.$$

$${\mu }_2={\mu }_2^{\prime }-{\mu }_1^{\prime 2}.$$

(23)

$${\mu }_2={\displaystyle \frac{e^{2\varphi }(2({\varphi }^4+\varphi )sinh\left(\varphi \right)+2{({\varphi }^3+1)}^{2}cosh\left(\varphi \right)-\left((2{\varphi }^4+4\varphi +1){\varphi }^2\right)-2)}{{(e^{\varphi }-1)}^4{({\varphi }^3+1)}^2}}.$$

(24)

$${\mu }_3={\mu }_3^{\prime }-3{\mu }_2^{\prime }{\mu }_1^{\prime }+2{\mu }_1^{\prime 3}.$$

$$\begin{array}{cc}\hfill {\mu }_3& ={\displaystyle \frac{1}{{(e^{\varphi }-1)}^6{({\varphi }^3+1)}^3}}{e}^{\varphi }(-\left(({\varphi }^3-\varphi +1){({\varphi }^3+1)}^2\right)+e^{4\varphi }({\varphi }^3-\varphi +1){({\varphi }^3+1)}^2\hfill \\ & -2e^{2\varphi }\varphi (3{\varphi }^6+6{\varphi }^3-{\varphi }^2+3)-e^{3\varphi }(2{\varphi }^9-2{\varphi }^7+6{\varphi }^6+3{\varphi }^5-4{\varphi }^4+6{\varphi }^3+3{\varphi }^2\hfill \\ & -2\varphi +2)e^{\varphi }(2{\varphi }^9+2{\varphi }^7+6{\varphi }^6+3{\varphi }^5+4{\varphi }^4+6{\varphi }^3+3{\varphi }^2+2\varphi +2)).\hfill \end{array}$$

(25)

$${\mu }_4={\mu }_3^{\prime }-4{\mu }_3^{\prime }\mu 1^{\prime }+6{\mu }_2^{\prime }{\mu }_1^{\prime 2}-3{\mu }_1^{\prime 4}.$$

$$\begin{array}{cc}\hfill {\mu }_4& ={\displaystyle \frac{1}{{(e^{\varphi }-1)}^8{({\varphi }^3+1)}^4}}(-3{({\varphi }^3-e^{\varphi }({\varphi }^3+\varphi +1)+1)}^4+6(e^{\varphi }-1)({\varphi }^3+1)(-{\varphi }^3\hfill \\ & +e^{2\varphi }({\varphi }^3+\varphi +1)+3e^{\varphi }\varphi -1){({\varphi }^3-e^{\varphi }({\varphi }^3+\varphi +1)+1)}^2+{(e^{\varphi }-1)}^3{({\varphi }^3+1)}^3\hfill \\ & (-{\varphi }^3+e^{4\varphi }({\varphi }^3+\varphi +1)-5e^{\varphi }(2{\varphi }^3-3\varphi +2)+5e^{3\varphi }(2{\varphi }^3+5\varphi +2)+55e^{2\varphi }\varphi -1)\hfill \\ & -4{(e^{\varphi }-1)}^2{({\varphi }^3+1)}^2(-{\varphi }^3+e^{\varphi }({\varphi }^3+\varphi +1)-1)(-{\varphi }^3+e^{\varphi }(-3{\varphi }^3+7\varphi -3)\hfill \\ & +e^{3\varphi }({\varphi }^3+\varphi +1)+e^{2\varphi }(3{\varphi }^3+10\varphi +3)-1\left)\right).\hfill \end{array}$$

(26)

Now, as is well known, the first original moment in Equation (15) represents the mean, and the second moment about the mean represents the variance, which can be obtained from Equation (23). Thus, the variance of the DLE distribution is given in Equation (24). Further, the coefficients of skewness and kurtosis of the DLE distribution can be calculated by substituting Equations (25) and (26) in the following formulas:

$$\begin{array}{cc}\hfill SK=& {\displaystyle \frac{{\mu }_3}{{\sigma }^3}}=(e^{\varphi }(-\left(({\varphi }^3-\varphi +1){({\varphi }^3+1)}^2\right)+e^{4\varphi }({\varphi }^3-\varphi +1){({\varphi }^3+1)}^2-2e^{2\varphi }\varphi (3{\varphi }^6\hfill \\ & +6{\varphi }^3-{\varphi }^2+3)-e^{3\varphi }(2{\varphi }^9-2{\varphi }^7+6{\varphi }^6+3{\varphi }^5-4{\varphi }^4+6{\varphi }^3+3{\varphi }^2-2\varphi +2)\hfill \\ & e^{\varphi }(2{\varphi }^9+2{\varphi }^7+6{\varphi }^6+3{\varphi }^5+4{\varphi }^4+6{\varphi }^3+3{\varphi }^2+2\varphi +2)\left)\right)/({(e^{\varphi }-1)}^6{({\varphi }^3+1)}^3\hfill \\ & {{\displaystyle \left(\frac{e^{2\varphi }(2({\varphi }^4+\varphi )sinh\left(\varphi \right)+2{({\varphi }^3+1)}^{2}cosh\left(\varphi \right)-\left((2{\varphi }^4+4\varphi +1){\varphi }^2\right)-2)}{{(e^{\varphi }-1)}^4{({\varphi }^3+1)}^2}\right)}}^{3/2}).\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill KU& ={\displaystyle \frac{{\mu }_4}{{\sigma }^4}}=(-3{({\varphi }^3-e^{\varphi }({\varphi }^3+\varphi +1)+1)}^4+6(e^{\varphi }-1)({\varphi }^3+1)(-{\varphi }^3+e^{2\varphi }({\varphi }^3\hfill \\ & +\varphi +1)+3e^{\varphi }\varphi -1){({\varphi }^3-e^{\varphi }({\varphi }^3+\varphi +1)+1)}^2+{(e^{\varphi }-1)}^3{({\varphi }^3+1)}^3(-{\varphi }^3\hfill \\ & +e^{4\varphi }({\varphi }^3+\varphi +1)-5e^{\varphi }(2{\varphi }^3-3\varphi +2)+5e^{3\varphi }(2{\varphi }^3+5\varphi +2)+55e^{2\varphi }\varphi -1)\hfill \\ & -4{(e^{\varphi }-1)}^2{({\varphi }^3+1)}^2(-{\varphi }^3+e^{\varphi }({\varphi }^3+\varphi +1)-1)(-{\varphi }^3+e^{\varphi }(-3{\varphi }^3+7\varphi -3)\hfill \\ & +e^{3\varphi }({\varphi }^3+\varphi +1)+e^{2\varphi }(3{\varphi }^3+10\varphi +3)-1\left)\right)/({(e^{\varphi }-1)}^8{({\varphi }^3+1)}^4\hfill \\ & {\left({\displaystyle \frac{e^{2\varphi }(2({\varphi }^4+\varphi )sinh\left(\varphi \right)+2{({\varphi }^3+1)}^{2}cosh\left(\varphi \right)-\left((2{\varphi }^4+4\varphi +1){\varphi }^2\right)-2)}{{(e^{\varphi }-1)}^4{({\varphi }^3+1)}^2}}\right)}^2).\hfill \end{array}$$

(28)

2.4. Coefficient of Variation (CV)

The CV is defined by the ratio of the standard deviation to the mean. The CV of the DLE distribution is given as:

$$CV={\displaystyle \frac{{(e^{\varphi }-1)}^2({\varphi }^3+1)\sqrt{\frac{csc{h}^4\left(\frac{\varphi }{2}\right)2({\varphi }^4+\varphi )sinh\left(\varphi \right)+2{({\varphi }^3+1)}^{2}cosh\left(\varphi \right)-\left((2{\varphi }^4+4\varphi +1){\varphi }^2\right)-2)}{{({\varphi }^3+1)}^2}}}{4e^{\varphi }({\varphi }^3+\varphi +1)-4({\varphi }^3+1)}}.$$

(29)

2.5. Dispersion Index

The DI is defined by the variance-to-mean ratio. The DI shows whether a distribution is suitable for modeling under-dispersed data or over-dispersed data. The DI of the DLE distribution is given as:

$$DI={\displaystyle \frac{e^{2\varphi }(2({\varphi }^4+\varphi )sinh\left(\varphi \right)+2{({\varphi }^3+1)}^{2}cosh\left(\varphi \right)-\left((2{\varphi }^4+4\varphi +1){\varphi }^2\right)-2)}{{(e^{\varphi }-1)}^2({\varphi }^3+1)(-{\varphi }^3+e^{\varphi }({\varphi }^3+\varphi +1)-1)}}.$$

(30)

Some numerical computations for mean, variance, skewness, kurtosis DI and CV based on DLE distribution parameters are listed in

Table 1. Mean, variance, skewness, kurtosis, DI and CV of the DLE distribution.

Parameter	Measures
ϕ	Mean	Variance	Skewness	Kurtosis	DI	CV
0.05	39.415	782.421	1.178	4.581	19.758	0.706
0.10	19.668	202.679	1.276	4.996	10.256	0.7199
0.20	9.415	50.197	1.290	5.165	5.306	0.748
0.30	6.095	22.251	1.296	5.125	3.6368	0.770
0.40	4.330	12.552	1.291	5.046	2.883	0.814
0.50	3.264	7.733	1.311	5.379	2.353	0.846
0.60	2.545	5.544	1.362	5.411	2.166	0.921
0.70	2.022	3.972	1.315	5.142	1.948	0.979
0.80	1.596	2.930	1.459	5.694	1.832	1.071
0.90	1.268	2.185	1.503	5.708	1.712	1.161
1.00	1.050	1.725	1.558	5.896	1.630	1.245
1.50	0.409	0.527	1.995	7.436	1.276	1.777

.

From Table 1 and Figure 3, it is concluded that as $\varphi$ increases, the mean and variance are decreased, while the skewness and kurtosis are increased. In addition, it is clear that the DLE distribution is suitable for modeling over-dispersed data sets since the DI is greater than one. The DLE distribution is also positively skewed with various shapes of kurtosis.

3. Parameter Estimation

In this section, classical and Bayesian estimation methods are considered to estimate the distribution parameter. In Section 3.1, the ML estimation method is discussed in detail. On the other hand, the Bayesian estimation method is discussed in Section 3.2. Two cases are investigated involving informative and non-informative priors based on various loss functions, including SELF, LINEX and GELF. A parameter interval estimation is considered in the two estimation methods.

3.1. ML Estimation

The method of ML is one of the most used classical point estimation techniques. Suppose $x_1,x_2,\dots ,x_n$ is a random sample of size n from the DLE distribution. Then, the likelihood function is given as:

$$L(\underline{x},\varphi )={\displaystyle \frac{e^{-n\varphi \overline{x}}}{{(1+{\varphi }^3)}^n}}\prod _{i=1}^n[(1+\varphi x_i+{\varphi }^3)-e^{-\varphi }(1+\varphi (x_i+1)+{\varphi }^3)].$$

(31)

The corresponding log-likelihood (LL) function is given as:

$$LL=-n\varphi \overline{x}-nlog(1+{\varphi }^3)+\sum _{i=1}^{n}log[(1+\varphi x_i+{\varphi }^3)-e^{-\varphi }(1+\varphi (x_i+1)+{\varphi }^3)].$$

(32)

By differentiating Equation (32) with respect to $\varphi$, we obtain the normal nonlinear likelihood equation as follows:

$${\displaystyle \frac{\partial LL}{\partial \varphi }}=-n\overline{x}-{\displaystyle \frac{3n\varphi }{(1+{\varphi }^3)}}+\sum _{i=1}^n{\displaystyle \frac{x_i+3{\varphi }^2+e^{-\varphi }(1+\varphi (x_i+1){\varphi }^3)-x_i-3{\varphi }^2}{[(1+\varphi x_i+{\varphi }^3)-e^{-\varphi }(1+\varphi (x_i+1)+{\varphi }^3)]}}.$$

(33)

Now, the ML estimator of parameter $\varphi$ can be found by equating Equation (33) to zero. Unfortunately, this equation cannot be solved explicitly; thus, numerical methods can be adopted to overcome this problem.

An asymptotic confidence interval (ACI) for $\varphi$ is constructed using the asymptotic distribution of ML of $\varphi$. We know the ML estimation $\varphi$ of $\widehat{\varphi }$ is consistent and asymptotic Gaussian distribution with $\sqrt{n}(\widehat{\varphi }-\varphi )\sim N(0,I^{-1}\left(\varphi \right))$, where $I^{-1}\left(\varphi \right)$ is the asymptotic variance of the ML estimator $\varphi$, which is the inverse of the observed Fisher information. The asymptotic observed Fisher information can be obtained as $I\left(\varphi \right)=E\left({\displaystyle \frac{{\partial }^{2}LL}{\partial {\varphi }^2}}\right)$, where the second-order partial derivative of the LL function is given as:

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^{2}LL}{\partial {\varphi }^2}}=& {\displaystyle \frac{-9n{\varphi }^4+6n\varphi (1+{\varphi }^3)}{{(1+{\varphi }^3)}^2}}+\sum _{i=1}^n[{\displaystyle \frac{{(x_i+3{\varphi }^2+e^{-\varphi }(\varphi (x_i+1)+{\varphi }^3)-x_i-3{\varphi }^2)}^2}{{((1+\varphi x_i+{\varphi }^3)-e^{-\varphi }(1+\varphi (x_i+1)+{\varphi }^3))}^2}}+\hfill \\ & {\displaystyle \frac{\left(e^{-\varphi }(\varphi (x_i+1)+{\varphi }^3-3{\varphi }^2-1)\right)(x_i+3{\varphi }^2+e^{-\varphi }(\varphi (x_i+1)+{\varphi }^3)-x_i-3{\varphi }^2)}{{((1+\varphi x_i+{\varphi }^3)-e^{-\varphi }(1+\varphi (x_i+1)+{\varphi }^3))}^2}}]\hfill \end{array}$$

(34)

Therefore, the $100\times (1-\gamma )\%$ ACI for the parameter $\varphi$ is:

$$\widehat{\varphi }\pm Z_{\gamma /2}\sqrt{v\left(\widehat{\varphi }\right)},$$

(35)

where $Z_{\gamma /2}$ is the upper $\gamma /2$ quantile of the standard Gaussian distribution, and $v\left(\widehat{\varphi }\right)=I^{-1}\left(\widehat{\varphi }\right)$, where $I\left(\widehat{\varphi }\right)=-{\displaystyle \frac{{\partial }^{2}LL}{{\varphi }^2}}{|}_{\varphi =\widehat{\varphi }}$.

3.2. Bayesian Estimation

The Bayesian estimation method is considered, involving both informative and non-informative prior functions based on different loss functions and associated credible intervals for the DLE distribution parameter.

• Case I: Informative Prior:
  Assume the parameter $\varphi$ is flowing in a gamma distribution with shape $\alpha$ and rate parameter 1. We use the gamma prior because of its advantages of flexibility and inclusiveness of several prior beliefs used by the researcher. The hyperparameter of the gamma prior was selected in such a way that the gamma prior mean (shape/rate) was the same as the original mean (parameter value); for more details, see [15,16]. The prior density function of parameters $\varphi$ is given by:

  $${\pi }_1(\varphi ,\alpha ,\beta )={\displaystyle \frac{}{\Gamma \left(\alpha \right)}}{\varphi }^{\alpha -1}{e}^{-\varphi };\varphi >0,\alpha >0,$$

  (36)

  where $\varphi$ is positive, and then the posterior density function of $\varphi$ given the data $x=(x_1,x_2\dots ,x_n)$ can be obtained as follows:

  $$f_1\left(\varphi \right|\underline{x})\propto {\displaystyle \frac{{\varphi }^{\alpha -1}{e}^{-\varphi }{e}^{-n\varphi \overline{x}}}{\Gamma \alpha {(1+{\varphi }^3)}^n}}\prod _{i=1}^n[(1+\varphi x_i+{\varphi }^3)-e^{-\varphi }(1+\varphi (x_i+1)+{\varphi }^3)].$$

  (37)
• Case II: Non-informative Prior:
  In this case, the unknown parameter has no or insufficient prior information. Assuming that the prior distribution of the parameter $\varphi$ follows a uniform distribution with PDF given by:

  $${\pi }_2\left(\varphi \right)={\displaystyle \frac{1}{\varphi }};\varphi >0,$$

  (38)

  and the posterior density function of $\varphi$ can be obtained as follows:

  $$f_2\left(\varphi \right|\underline{x})\propto {\displaystyle \frac{e^{-n\varphi \overline{x}}}{\varphi {(1+{\varphi }^3)}^n}}\prod _{i=1}^n[(1+\varphi x_i+{\varphi }^3)-e^{-\varphi }(1+\varphi (x_i+1)+{\varphi }^3)].$$

  (39)

For the above two cases, the Bayesian estimator of $\varphi$ is obtained under three types of loss functions as mentioned in the following section (Section 3.2.1, Section 3.2.2 and Section 3.2.3).

3.2.1. Bayesian Estimation under SELF

The following is a commonly used loss function and is classified as a symmetric function. The SELF is defined as:

$$SE(\widehat{\varphi },\varphi )={(\widehat{\varphi }-\varphi )}^2.$$

Therefore, the Bayes estimate under the SELF is the mean of the posterior distribution, which can be written for two cases as follows:

$${\widehat{\varphi }}_{SELF}\propto {\int }_{\varphi }\varphi f\left(\varphi \right|\underline{x})d\varphi .$$

$$\begin{array}{cc}\hfill {\widehat{{\varphi }_1}}_{SELF}& \propto {\int }_0^{\infty }{\displaystyle \frac{{\varphi }^{\alpha }{e}^{-\varphi }{e}^{-n\varphi \overline{x}}}{\Gamma \alpha {(1+{\varphi }^3)}^n}}\prod _{i=1}^n[(1+\varphi x_i+{\varphi }^3)-e^{-\varphi }(1+\varphi (x_i+1)+{\varphi }^3)]d\varphi .\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill {\widehat{{\varphi }_2}}_{SELF}& \propto {\int }_0^{\infty }{\displaystyle \frac{e^{-n\varphi \overline{x}}}{{(1+{\varphi }^3)}^n}}\prod _{i=1}^n[(1+\varphi x_i+{\varphi }^3)-e^{-\varphi }(1+\varphi (x_i+1)+{\varphi }^3)]d\varphi .\hfill \end{array}$$

(41)

3.2.2. Bayesian Estimation under LINEX Loss Function

The LINEX loss function, introduced by [17], is considered as an asymmetric loss function. The LINEX loss function is derived as follows:

$$LIN(\widehat{\varphi },\varphi )=(e^{c(\widehat{\varphi }-\varphi )}-c(\widehat{\varphi }-\varphi )-1),$$

where a shape parameter $c\ne 0$ reflects the direction and degree of asymmetry. Therefore, when $c>1$, it implies that an overestimation is more serious than an underestimation, and vice versa for $c<0$. Further, the LINEX loss function will be close to SELF when c is near to zero. The Bayesian estimator of $\varphi$ under a LINEX loss function is as follows:

$$\widehat{{\varphi }_{LIN}}\propto {\displaystyle \frac{1}{c}}ln\left[{\int }_{\varphi }{e}^{-c\varphi }f\left(\varphi \right|\underline{x})d\varphi \right].$$

$${\widehat{{\varphi }_1}}_{LIN}\propto -{\displaystyle \frac{1}{c}}ln\left[{\int }_0^{\infty }{\displaystyle \frac{e^{-c\varphi }{\varphi }^{\alpha -1}{e}^{-\varphi }{e}^{-n\varphi \overline{x}}}{\Gamma \alpha {(1+{\varphi }^3)}^n}}\prod _{i=1}^n[(1+\varphi x_i+{\varphi }^3)-e^{-\varphi }(1+\varphi (x_i+1)+{\varphi }^3)]d\varphi \right].$$

(42)

$${\widehat{{\varphi }_2}}_{LIN}\propto -{\displaystyle \frac{1}{c}}ln\left[{\int }_0^{\infty }{\displaystyle \frac{e^{-c\varphi }{e}^{-n\varphi \overline{x}}}{\varphi {(1+{\varphi }^3)}^n}}\prod _{i=1}^n[(1+\varphi x_i+{\varphi }^3)-e^{-\varphi }(1+\varphi (x_i+1)+{\varphi }^3)]d\varphi \right].$$

(43)

3.2.3. Bayesian Estimation under GELF

The GELF was proposed by [18]. The GELF is an asymmetric loss function and can be defined as

$$GE(\widehat{\varphi },\varphi )={\displaystyle \frac{q^2}{2}}{(ln\widehat{\varphi }-ln\varphi )}^2,$$

where q is a shape parameter that denotes the direction and degree of asymmetry. It implies that an overestimation is more significant than underestimating when $q>0$ and vice versa when $q<0$. The Bayesian estimator of $\varphi$ under GELF is as follows:

$$\widehat{{\varphi }_{GELF}}\propto {\left[{\int }_{\varphi }{\varphi }^{-q}f\left(\varphi \right|\underline{x})d\varphi \right]}^{-1/q}.$$

$${\widehat{{\varphi }_1}}_{GELF}\propto {\left[{\int }_0^{\infty }{\displaystyle \frac{{\varphi }^{-q}{\varphi }^{\alpha -1}{e}^{-\varphi }{e}^{-n\varphi \overline{x}}}{\Gamma \alpha {(1+{\varphi }^3)}^n}}\prod _{i=1}^n[(1+\varphi x_i+{\varphi }^3)-e^{-\varphi }(1+\varphi (x_i+1)+{\varphi }^3)]d\varphi \right]}^{-1/q}.$$

(44)

$${\widehat{{\varphi }_2}}_{GELF}\propto {\left[{\int }_0^{\infty }{\displaystyle \frac{{\varphi }^{-q}{e}^{-n\varphi \overline{x}}}{\varphi {(1+{\varphi }^3)}^n}}\prod _{i=1}^n[(1+\varphi x_i+{\varphi }^3)-e^{-\varphi }(1+\varphi (x_i+1)+{\varphi }^3)]d\varphi \right]}^{-1/q}.$$

(45)

The Bayesian estimator of $\varphi$ for different loss functions cannot be achieved in closed forms, as is evident from Equations (40)–(45). This can be found numerically using Bayes MCMC techniques to generate samples from the posterior distribution and then obtain the Bayesian estimators of $\varphi$ along with three different loss functions.

The 100 (1 − $\gamma$) % credible intervals are constructed by using the following formula:

$$P(LL<\varphi <UL)={\int }_{LL}^{UL}f\left(\varphi \right|\underline{X})d\varphi =1-\gamma ,0<\gamma <1,$$

(46)

where LL and UL are the lower and upper limits of the 100 (1 − $\gamma$) % credible interval for $\varphi$, respectively, and $f\left(\varphi \right|\underline{X})$ is the posterior distribution of $\varphi$ given the sample $\underline{X}$. Since Equations (37) and (39) are not similar to any well-known distributions, we use plots of these posterior distributions to show that the distributions in Equations (37) and (39) behave similarly to the normal distribution, see Figure 4. Thus, we employ the Metropolis-Hastings technique with normal proposal distributions, as follows:

• step 1: Set the initial values ${\varphi }^{\left(0\right)}=\widehat{\varphi }.$
• step 2: Set i = 1.
• step 3: Generate ${\varphi }^{(*)}$ from $q\left(\varphi \right)=N(\widehat{\varphi },\widehat{Var}\left(\widehat{\varphi }\right)).$
• step 4: Obtain $h=min(1,{\displaystyle \frac{f\left({\varphi }^{*}\right|\underline{\mathbf{x}})}{f\left({\varphi }^{(i-1)}\right|\underline{\mathbf{x}})}}).$
• step 5: Generate sample U from the uniform U(0,1) distribution.
• step 6: if $U\le h$, then set ${\varphi }^{\left(i\right)}={\varphi }^{*}$; otherwise ${\varphi }^{\left(i\right)}={\varphi }^{i-1}.$
• step 7: Set i = i + 1.
• step 8: Repeat steps 2–7, M times, and obtain ${\varphi }^{\left(i\right)},i=1,\dots ,M$.
• step 9: Under SELF, obtain the Bayes estimates of $\widehat{\varphi }$ as:

  $${\widehat{\varphi }}_{SELF}=\sum _{i=Q+1}^M{\displaystyle \frac{{\varphi }^{\left(i\right)}}{M-Q}},$$

  where Q is the burn-in period.
• step 10: To obtain the credible intervals of $\varphi$ using the algorithm proposed by [19] order ${\varphi }^{(Q+i)}$ as $({\varphi }^{[Q+i]},{\varphi }^{[Q+2]},\dots ,{\varphi }^{\left[M\right]}$ Then, the $100(1-\alpha )\%$ symmetric credible intervals of $\varphi$ becomes $({\varphi }^{\left[\right(\alpha /2)M-Q]},{\varphi }^{\left[\right(1-\alpha /2)M-Q]}).$
• step 11: Under LINEX loss function, obtain the Bayes estimates of $\widehat{\varphi }$ as:

  $${\widehat{\varphi }}_{LIN}={\displaystyle \frac{-1}{c}}ln[\sum _{i=Q+1}^M{e}^{-c{\varphi }^{\left(i\right)}}/M-Q].$$
• step 12: Under GELF, obtain the Bayes estimates of $\widehat{\varphi }$ as:

  $${\widehat{\varphi }}_{GE}={[\sum _{Q+i=1}^M{\varphi }^{{\left(i\right)}^{-q}}/M-Q]}^{-1/q}.$$

4. Simulation

In this section, a simulation study is conducted to investigate the performance of the unknown estimator based on 1000 samples generated from the DLE distribution at different sample sizes (n = 20, 100, 500, 1000) and four choices of $\varphi =0.01,0.1,0.5,1$. The ML estimator is computed and a 95% ACI for $\varphi$ is constructed. Additionally, the corresponding Bayesian estimates based on various loss functions are calculated using different priors along with 95% credible intervals. The MCMC algorithm is adopted to generate the 10000 MCMC samples, after discarding the first 2000 values as a burn-in period, and convergence is checked. A comparison is made between the ML estimate and its corresponding Bayesian estimates for each case mentioned in Section 3.2 to assess their performances. The ML and Bayesian estimates using three different loss functions, bias, root mean squared error (RMSE) and the length of 95% ACI and credible intervals are shown in

Table 2. The Classical and Bayes estimates with different loss functions of $\varphi$ for Case I: Gamma prior.

ϕ	n	ML (Bias) (RMSE) (Length)	Bayesian Estimation
			SELF (Bias) (RMSE) (Length)	LINEX (Bias) (RMSE) (Length)		GELF (Bias) (RMSE) (Length)
				c = −1.5	c = 1.5	q = −1.5	q = 1.5
0.01	20	1.022e-2 2.189e-4 1.654e-3 5.379e-3	1.024e-2 2.449e-4 1.708e-3 6.578e-3	1.026e-2 2.637e-4 1.711e-3 6.552e-3	1.026e-2 2.597e-4 1.709e-3 6.547e-3	1.033e-2 3.255e-4 1.731e-3 6.588e-3	9.941e-3 −5.934e-5 1.638e-3 6.351e-3
	100	1.009e-2 9.240e-5 7.238e-4 2.356e-3	1.005e-2 4.762e-5 7.297e-4 2.933e-3	1.001e-2 1.016e-5 7.155e-4 2.832e-3	1.001e-2 9.408e-6 7.154e-4 2.832e-3	1.002e-2 2.225e-5 7.166e-4 2.835e-3	9.947e-3 −5.265e-5 7.129e-4 2.816e-3
	500	1.000e-2 2.500e-6 3.115e-4 1.022e-3	1.001e-2 8.865e-6 3.131e-4 1.210e-3	1.002e-2 2.388e-5 3.220e-4 1.254e-3	1.002e-2 2.373e-5 3.220e-4 1.254e-3	1.003e-2 2.631e-5 3.223e-4 1.255e-3	1.001e-2 1.129e-5 3.209e-4 1.252e-3
	1000	1.001e-2 9.906e-6 2.277e-4 7.465e-4	1.001e-2 8.347e-6 2.326e-4 9.391e-4	1.000e-2 3.193e-6 2.274e-4 8.756e-4	1.000e-2 3.118e-6 2.274e-4 8.756e-4	1.000e-2 4.404e-6 2.274e-4 8.758e-4	9.997e-3 −3.089e-6 2.273e-4 8.748e-4
0.1	20	1.014e-1 1.443e-3 1.684e-2 5.506e-2	1.020e-1 1.997e-3 1.672e-2 6.330e-2	1.034e-1 3.433e-3 1.710e-2 6.345e-2	1.030e-1 3.025e-3 1.690e-2 6.296e-2	1.033e-1 3.253e-3 1.731e-2 6.547e-2	1.000e-1 1.536e-5 1.618e-2 6.134e-2
	100	1.002e-1 2.479e-4 6.984e-3 2.290e-2	1.004e-1 3.646e-4 7.272e-3 2.859e-2	1.007e-1 7.437e-4 7.145e-3 2.757e-2	1.007e-1 6.675e-4 7.127e-3 2.753e-2	1.002e-1 2.160e-4 7.161e-3 2.837e-2	1.001e-1 7.786e-5 7.057e-3 2.741e-2
	500	9.999e-2 −1.212e-5 3.153e-3 1.035e-2	1.002e-1 1.711e-4 3.103e-3 1.247e-2	1.000e-1 4.091e-5 3.173e-3 1.242e-2	1.000e-1 2.594e-5 3.172e-3 1.241e-2	1.003e-1 2.611e-4 3.216e-3 1.248e-2	9.991e-2 −9.114e-5 3.170e-3 1.239e-2
	1000	9.997e-2 −3.075e-5 2.253e-3 7.392e-3	1.000e-1 −4.288e-8 2.237e-3 8.769e-3	1.000e-1 2.439e-5 2.213e-3 8.683e-3	1.000e-1 1.691e-5 2.212e-3 8.682e-3	1.000e-1 4.638e-5 2.272e-3 8.747e-3	9.996e-2 −4.157e-5 2.211e-3 8.675e-3
0.5	20	5.044e-1 4.449e-3 7.286e-2 2.386e-1	5.084e-1 8.356e-3 7.386e-2 2.966e-1	5.111e-1 1.107e-2 7.600e-2 2.976e-1	5.030e-1 3.038e-3 7.328e-2 2.894e-1	5.091e-1 9.129e-3 7.501e-2 2.840e-1	4.939e-1 −6.122e-3 7.318e-2 2.875e-1
	100	4.993e-1 -6.802e-4 3.228e-2 1.059e-1	5.031e-1 3.059e-3 3.202e-2 1.239e-1	5.012e-1 1.186e-3 3.131e-2 1.221e-1	4.996e-1 −3.863e-4 3.113e-2 1.213e-1	4.998e-1 -1.744e-4 3.254e-2 1.255e-1	4.978e-1 −2.218e-3 3.118e-2 1.211e-1
	500	5.009e-1 8.547e-4 1.452e-2 4.757e-2	5.011e-1 1.075e-3 1.465e-2 5.646e-2	5.009e-1 8.615e-4 1.419e-2 5.472e-2	5.005e-1 5.468e-4 1.416e-2 5.469e-2	5.011e-1 1.113e-3 1.479e-2 5.767e-2	5.002e-1 1.804e-4 1.415e-2 5.471e-2
	1000	4.998e-1 −2.207e-4 1.024e-2 3.358e-2	5.003e-1 3.112e-4 1.010e-2 4.018e-2	5.002e-1 2.365e-4 1.016e-2 3.914e-2	5.001e-1 7.945e-5 1.015e-2 3.913e-2	5.001e-1 9.780e-5 1.042e-2 4.037e-2	4.999e-1 −1.037e-4 1.015e-2 3.914e-2
1	20	1.033e+0 3.251e-2 1.526e-1 4.892e-1	1.022e+0 2.247e-2 1.656e-1 6.310e-1	1.042e+0 4.163e-2 1.630e-1 6.061e-1	1.010e+0 1.011e-2 1.451e-1 5.574e-1	1.029e+0 2.931e-2 1.540e-1 5.656e-1	1.001e+0 9.998e-4 1.453e-1 5.589e-1
	100	1.007e+0 6.967e-3 6.140e-2 2.002e-1	1.005e+0 4.674e-3 6.061e-2 2.436e-1	1.009e+0 8.762e-3 6.223e-2 2.432e-1	1.003e+0 3.126e-3 6.085e-2 2.402e-1	1.007e+0 7.072e-3 6.218e-2 2.408e-1	1.001e+0 1.297e-3 6.081e-2 2.403e-1
	500	1.002e+0 1.674e-3 2.682e-2 8.785e-2	1.002e+0 2.104e-3 2.757e-2 1.071e-1	1.002e+0 1.542e-3 2.856e-2 1.099e-1	1.000e+0 4.389e-4 2.845e-2 1.096e-1	1.001e+0 9.337e-4 2.827e-2 1.126e-1	1.000e+0 7.295e-5 2.844e-2 1.096e-1
	1000	1.000e+0 2.484e-4 1.876e-2 6.154e-2	1.000e+0 3.921e-4 1.969e-2 7.778e-2	1.001e+0 8.097e-4 1.933e-2 7.923e-2	1.000e+0 2.593e-4 1.929e-2 7.908e-2	9.996e-1 −4.284e-4 1.929e-2 7.516e-2	1.000e+0 7.631e-5 1.928e-2 7.907e-2

and

Table 3. The Classical and Bayes estimates with different loss functions of $\varphi$ for Case II: Uniform prior.

$\mathit{\varphi }$	n	ML (Bias) (RMSE) (Length)	Bayesian Estimation
			SELF (Bias) (RMSE) (Length)	LINEX (Bias) (RMSE) (Length)		GELF (Bias) (RMSE) (Length)
				c = −1.5	c = 1.5	q = −1.5	q = 1.5
0.01	20	1.022e-2 2.189e-4 1.654e-3 5.379e-3	1.052e-2 5.208e-4 1.809e-3 6.725e-3	1.052e-2 5.208e-4 1.809e-3 6.725e-3	1.052e-2 5.166e-4 1.806e-3 6.720e-3	1.058e-2 5.826e-4 1.836e-3 6.764e-3	1.020e-2 1.975e-4 1.690e-3 6.515e-3
	100	1.009e-2 9.240e-5 7.238e-4 2.356e-3	1.006e-2 6.049e-5 7.218e-4 2.860e-3	1.006e-2 6.049e-5 7.218e-4 2.860e-3	1.006e-2 5.974e-5 7.216e-4 2.859e-3	1.007e-2 7.258e-5 7.237e-4 2.862e-3	9.998e-3 −2.285e-6 7.147e-4 2.846e-3
	500	1.000e-2 2.500e-6 3.115e-4 1.022e-3	1.003e-2 3.384e-5 3.232e-4 1.253e-3	1.003e-2 3.384e-5 3.232e-4 1.253e-3	1.003e-2 3.369e-5 3.232e-4 1.253e-3	1.004e-2 3.627e-5 3.236e-4 1.254e-3	1.002e-2 2.125e-5 3.218e-4 1.252e-3
	1000	1.001e-2 9.906e-6 2.277e-4 7.465e-4	1.001e-2 8.132e-6 2.277e-4 8.731e-4	1.001e-2 8.132e-6 2.277e-4 8.731e-4	1.001e-2 8.057e-6 2.277e-4 8.731e-4	1.001e-2 9.344e-6 2.277e-4 8.732e-4	1.000e-2 1.846e-6 2.274e-4 8.722e-4
0.1	20	1.014e-1 1.443e-3 1.684e-2 5.506e-2	1.045e-1 4.538e-3 1.760e-2 6.540e-2	1.060e-1 6.007e-3 1.819e-2 6.480e-2	1.056e-1 5.588e-3 1.793e-2 6.431e-2	1.064e-1 6.437e-3 1.837e-2 6.490e-2	1.026e-1 2.579e-3 1.679e-2 6.267e-2
	100	1.002e-1 2.479e-4 6.984e-3 2.290e-2	1.009e-1 8.665e-4 7.351e-3 2.888e-2	1.012e-1 1.243e-3 7.248e-3 2.768e-2	1.012e-1 1.166e-3 7.224e-3 2.764e-2	1.013e-1 1.330e-3 7.267e-3 2.770e-2	1.006e-1 5.769e-4 7.114e-3 2.750e-2
	500	9.999e-2 −1.212e-5 3.153e-3 1.035e-2	1.003e-1 2.695e-4 3.113e-3 1.248e-2	1.001e-1 1.405e-4 3.179e-3 1.246e-2	1.001e-1 1.255e-4 3.178e-3 1.246e-2	1.002e-1 1.579e-4 3.180e-3 1.246e-2	1.000e-1 8.480e-6 3.172e-3 1.244e-2
	1000	9.997e-2 −3.075e-5 2.253e-3 7.392e-3	1.001e-1 5.056e-5 2.239e-3 8.759e-3	1.001e-1 7.373e-5 2.215e-3 8.686e-3	1.001e-1 6.625e-5 2.215e-3 8.685e-3	1.001e-1 8.244e-5 2.216e-3 8.687e-3	1.000e-1 7.726e-6 2.213e-3 8.679e-3
0.5	20	5.044e-1 4.449e-3 7.286e-2 2.386e-1	5.189e-1 1.887e-2 7.669e-2 2.982e-1	5.216e-1 2.163e-2 7.923e-2 3.040e-1	5.134e-1 1.340e-2 7.540e-2 2.953e-1	5.201e-1 2.008e-2 7.808e-2 3.007e-1	5.043e-1 4.292e-3 7.403e-2 2.938e-1
	100	4.993e-1 -6.802e-4 3.228e-2 1.059e-1	5.052e-1 5.174e-3 3.238e-2 1.238e-1	5.033e-1 3.291e-3 3.154e-2 1.219e-1	5.017e-1 1.711e-3 3.125e-2 1.213e-1	5.030e-1 3.022e-3 3.145e-2 1.216e-1	4.999e-1 −1.200e-4 3.118e-2 1.211e-1
	500	5.009e-1 8.547e-4 1.452e-2 4.757e-2	5.015e-1 1.490e-3 1.470e-2 5.639e-2	5.013e-1 1.291e-3 1.423e-2 5.472e-2	5.010e-1 9.755e-4 1.419e-2 5.469e-2	5.012e-1 1.238e-3 1.422e-2 5.470e-2	5.006e-1 6.089e-4 1.417e-2 5.471e-2
	1000	4.998e-1 −2.207e-4 1.024e-2 3.358e-2	5.005e-1 5.209e-4 1.011e-2 4.015e-2	5.004e-1 4.443e-4 1.017e-2 3.920e-2	5.003e-1 2.872e-4 1.015e-2 3.918e-2	5.004e-1 4.181e-4 1.016e-2 3.919e-2	5.001e-1 1.041e-4 1.015e-2 3.919e-2
1	20	1.033e+0 3.251e-2 1.526e-1 4.892e-1	1.043e+0 4.315e-2 1.756e-1 6.534e-1	1.063e+0 6.317e-2 1.755e-1 6.263e-1	1.030e+0 2.974e-2 1.522e-1 5.732e-1	1.051e+0 5.099e-2 1.653e-1 6.028e-1	1.021e+0 2.067e-2 1.513e-1 5.752e-1
	100	1.007e+0 6.967e-3 6.140e-2 2.002e-1	1.008e+0 8.408e-3 6.131e-2 2.448e-1	1.013e+0 1.253e-2 6.320e-2 2.458e-1	1.007e+0 6.832e-3 6.148e-2 2.423e-1	1.011e+0 1.060e-2 6.250e-2 2.444e-1	1.005e+0 5.001e-3 6.132e-2 2.424e-1
	500	1.002e+0 1.674e-3 2.682e-2 8.785e-2	1.003e+0 2.831e-3 2.764e-2 1.070e-1	1.002e+0 2.278e-3 2.865e-2 1.101e-1	1.001e+0 1.173e-3 2.850e-2 1.098e-1	1.002e+0 1.908e-3 2.859e-2 1.100e-1	1.001e+0 8.077e-4 2.849e-2 1.098e-1
	1000	1.000e+0 2.484e-4 1.876e-2 6.154e-2	1.001e+0 7.598e-4 1.971e-2 7.810e-2	1.001e+0 1.194e-3 1.937e-2 7.903e-2	1.001e+0 6.434e-4 1.931e-2 7.891e-2	1.001e+0 1.010e-3 1.935e-2 7.898e-2	1.000e+0 4.606e-4 1.931e-2 7.890e-2

. All computational algorithms were implemented using the R-package.

Table 2 and Table 3 show the results of the suggested techniques for calculating the parameter estimates. Based on the reported tables, the following observations are permissible:

• The RMSE of ML estimates and Bayes estimates for different loss function of $\varphi$ decrease as the sample size increases.
• The estimates are asymptotically unbiased since they are more accurate as the sample size increases.
• The parameter estimates come from the best unbiased estimator when the RMSE value is near zero.
• The RMSE and length of a credible interval for the Bayesian estimates with positive weight for the asymmetric loss function are smaller than the Bayesian estimates with negative weight for the asymmetric loss function.
• A GELF with a positive weight is better than the other loss functions.
• Bayesian estimation under GELF with positive weight is better than ML estimation for all sample sizes.

Figure 5 shows heat-map descriptions for the RMSE results in two cases, where 1 is for the case of the gamma prior, 2 is for the case of the uniform prior, P is the positive weight and N is the negative weight. The bold color represents the highest values of RMSE and the white color represents the lowest values of RMSE. Figure 5 supports the results of Table 2 and Table 3. We note that all methods have the same efficiency.

5. Applications

5.1. Real Data Modeling for Comparing the Competitive Discrete Models

In this section, three real datasets are used to demonstrate the efficiency of the proposed distribution. Moreover, the maximum log-likelihood (-LL) and the goodness-of-fit criteria, including the KS test, the corresponding p-value, AIC and BIC, are used to confirm that the DLE distribution has the best fit against some other competitive one-parameter discrete distributions, which are listed in

Table 4. The competitive models of the DLE distribution.

Models	Abbreviation	Author(s)
Discrete Raleigh	DR	[20]
Poisson	Pois	[21]
Discrete Pareto	DP	[22]
Discrete Burr-Hatke	DBH	[23]
Discrete Inverted Topp-Leone	DITL	[24]
Geometric	GEOM	[25]
Negative Binomial	Nbinom	[26]

.

5.1.1. Dataset I

Various methods are analyzed that evaluate the lifetime of electrical components, including maintenance records for electrical components, and methods such as modified open circuit voltage decay, small parallel resistance, and pulse recovery technique that assess carrier lifetime at the semiconductor–dielectric interface [27]. A sample of the failure time of 15 electronic components in an acceleration life test was used [28]. The mean and variance of the first dataset are 27.53 and 431.98, respectively. The dispersion index value is 15.689, which indicates that the dataset is over-dispersed.

Table 5. ML estimates, -LL and goodness-of-fit criteria of Dataset I.

Models	ML (S.E.)	-LL	AIC	BIC	K-S	p-Value
DLE	0.071 (0.013)	64.787	131.575	132.283	0.114	0.976
DR	24.382 (3.148)	66.394	134.79	135.50	0.2160	0.430
Pois	27.533 (1.355)	151.21	304.41	305.12	0.3810	0.018
DITL	0.4178 (0.107)	74.491	150.98	151.69	0.3590	0.031
DP	0.3284 (0.084)	77.402	156.80	157.51	0.4060	0.009
DBH	0.9992 (0.008)	91.368	184.74	185.44	0.7910	0.000
Geom	0.035 (0.009)	65.00	132.00	132.71	0.1768	0.673
Nbinom	0.3526 (0.018)	88.557	179.11	179.82	0.3087	0.091

introduces the ML estimator with its S.E. for the parameter of the DLE distribution and other competitive models. Moreover, the goodness-of-fit criteria are shown.

Regarding Table 5, it is clear that the distribution DR works quite well for analyzing these data aside from the DLE distribution. However, the DLE distribution is the best distribution among all the competitive distributions. Figure 6 supports the results of Table 5.

5.1.2. Dataset II

A sample of a 42-day daily new deaths COVID-19 dataset from the United States Virgin Islands, recorded between 19 April 2021 and 30 May 2021. These data are daily new deaths. The data are as follows: 11, 2, 3, 10, 10, 4, 12, 0, 10, 3, 5, 12, 6, 9, 13, 4, 10, 26, 0, 32, 0, 0, 13, 10, 3, 20, 5, 6, 0, 3, 18, 2, 18, 14, 24, 7, 0, 30, 16, 26, 17, 23. The data are available on the Worldometer website at [29]. The mean and variance of this dataset are 10.405 and 78.003, respectively. The dispersion index value is 7.497, which indicates that the dataset is over-dispersed.

Table 6. ML Estimates, -LL and goodness-of-fit criteria of Dataset II.

Models	ML (S.E.)	-LL	AIC	BIC	K-S	p-Value
DLE	0.184 (0.02)	147.79	297.59	299.32	0.127	0.508
DR	9.874 (0.762)	155.81	313.62	315.36	0.213	0.04
Pois	10.405 (0.498)	240.13	482.26	483.99	0.328	0.000
DITL	0.647 (0.099)	157.03	316.06	317.80	0.293	0.001
DP	0.472 (0.073)	162.72	327.44	329.18	0.342	0.000
DBH	0.995 (0.000)	177.99	357.97	359.71	0.614	0.000
GEOM	0.088 (0.013)	142.33	286.66	288.40	0.159	0.236
Nbinom	0.801 (0.009)	214.967	431.93	433.67	0.307	0.0007

introduces the ML estimator with its standard error for the parameter of DLE distribution and other competitive distributions. Moreover, the goodness-of-fit criteria are shown.

From Table 6, the DLE distribution is the best distribution among all competitive distributions except geometric distribution. Figure 7 supports the results of Table 6.

5.1.3. Dataset III

A sample of a 66-day daily new deaths COVID-19 dataset from China, recorded between January 23,2020, and March 28,2020. The data represent daily new deaths. The data are as follows: 8, 16, 15, 24, 26, 26, 38, 43, 46, 45, 57, 64, 65, 73, 73, 86, 89, 97, 108, 97, 146, 121, 143, 142, 105, 98, 136, 114, 118, 109, 97, 150, 71, 52, 29, 44, 47, 35, 42, 31, 38, 31, 30, 28, 27, 22, 17, 22, 11, 7, 13, 10, 14, 13, 11, 8, 3, 7, 6, 9, 7, 4, 6, 5, 3 and 5. The data are available on the Worldometer website at [29]. The mean and variance of this dataset are 47.742 and 1924.8 respectively. The dispersion index value is 38.696 which indicates that the dataset is over-dispersed.

Table 7. ML estimates, -LL and goodness-of-fit criteria of Dataset III.

Models	ML (S.E.)	-LL	AIC	BIC	K-S	p-Value
DLE	0.039 (0.003)	330.51	663.027	665.217	0.172	0.04
DR	47.010 (2.893)	347.23	696.455	698.644	0.293	0.000
Pois	49.742 (0.868)	1409.8	2821.565	2823.754	0.497	0.000
DITL	0.354 (0.044)	366.907	735.815	738.004	0.329	0.000
DP	0.286 (0.035)	379.070	760.14	762.33	0.382	0.000
DBH	0.999 (0.002)	461.02	924.04	926.23	0.812	0.000
GEOM	0.019 (0.002)	324.51	651.02	653.21	0.085	0.726
Nbinom	0.570 (0.006)	918.41	1838.81	1841.00	0.483	0.000

introduces the ML estimator with its S.E. for the parameter of DLE distribution and other competitive distributions. Moreover, the goodness-of-fit criteria are shown.

From Table 7, the DLE distribution is the best distribution among all competitive distributions. Figure 8 supports the results of Table 7.

5.2. Real Data Modeling for Comparing Classical and Bayesian Estimation Methods

In this section, we analyze data sets presented in previous sections to compare the classical and Bayesian estimation methods. We consider the AIC, BIC, KS tests and the corresponding p-value statistics for comparing the estimation methods.

5.2.1. Dataset I

The estimators under classical and Bayesian methods, AIC, BIC, KS tests and the p-value statistics for Dataset I are given in

Table 8. Estimators under classical and Bayesian estimation methods and goodness-of-fit criteria of Dataset I.

	Method	$\widehat{\mathit{\varphi }}$	AIC	BIC	K-S	p-Value
	ML	0.071	131.5754	132.2834	0.11441	0.9766
Case I	SELF	0.0709	131.5765	132.2846	0.112	0.981
	P-LINEX	0.0708	131.5773	132.2854	0.111	0.982
	N-LINEX	0.0710	131.5759	132.284	0.1126	0.9798
	P-GELF	0.0678	131.6501	132.3582	0.1172	0.9707
	N-GELF	0.0714	131.5755	132.2836	0.1152	0.9749
Case II	SELF	0.0735	131.6036	132.3116	0.12616	0.9464
	P-LINEX	0.0734	131.6003	132.3084	0.1255	0.9485
	N-LINEX	0.0737	131.607	132.3151	0.1268	0.9441
	P-GELF	0.076	131.5788	132.2869	0.1115	0.9817
	N-GELF	0.0741	131.6206	132.3287	0.1292	0.9361

.

As Table 8 shows, the classical estimation method ML is the best among all the Bayesian estimation methods used in our study. However, all other methods perform very well.

5.2.2. Dataset II

As

Table 9. Estimators under classical and Bayesian estimation methods and goodness-of-fit criteria of Dataset II.

	Method	$\widehat{\mathit{\varphi }}$	AIC	BIC	K-S	p-Value
	ML	0.1838	297.5856	299.3233	0.1269	0.5076
Case I	SELF	0.1834	297.5837	299.321	0.1270	0.5068
	P-LINEX	0.1831	297.5846	299.3223	0.1271	0.5063
	N-LINEX	0.1837	297.5833	299.3209	0.1269	0.5074
	P-GELF	0.1807	297.6085	299.3461	0.1275	0.5021
	N-GELF	0.1839	297.5832	299.3209	0.1269	0.5078
Case II	SELF	0.1854	297.5891	299.3268	0.1299	0.4772
	P-LINEX	0.1851	297.5871	299.3247	0.1291	0.4861
	N-LINEX	0.1857	297.5917	299.3294	0.1308	0.4684
	P-GELF	0.1827	297.5866	299.3243	0.1272	0.5056
	N-GELF	0.1859	297.594	299.3317	0.1315	0.4616

shows, the Bayesian estimation under GELF with negative weight for case I is the best among all the estimation methods used in our study. However, all other methods perform very well.

5.2.3. Dataset III

According to

Table 10. Estimators under classical and Bayesian estimation methods and goodness-of-fit criteria of Dataset III.

	Method	$\widehat{\mathit{\varphi }}$	AIC	BIC	K-S	p-Value
	ML	0.0398	663.027	665.2167	0.1718	0.0407
Case I	SELF	0.0397	663.0275	665.2172	0.1723	0.0398
	P-LINEX	0.0397	663.0277	665.2173	0.1723	0.0397
	N-LINEX	0.0397	663.0274	665.2171	0.1722	0.0399
	P-GELF	0.0393	663.0447	665.2344	0.1746	0.0357
	N-GELF	0.0398	663.027	665.2167	0.1718	0.0407
Case II	SELF	0.0401	663.0325	665.2221	0.1701	0.0438
	P-LINEX	0.0400	663.0321	665.2217	0.1701	0.0437
	N-LINEX	0.0401	663.0329	665.2225	0.1700	0.0439
	P-GELF	0.0397	663.0282	665.2179	0.1725	0.0394
	N-GELF	0.0401	663.0361	665.2258	0.1697	0.0448

, although the best performance among the estimation methods cannot be determined in case II, the classical ML and the Bayesian estimation under GELF with negative weight performs better than other methods according to all criteria in case I.

Figure 9, Figure 10 and Figure 11 display the trace, ACF, and density function DLE parameter values to check the convergence of the MCMC method after burn-in. These results imply that the chains converge well and that the burn-in period is appropriate.

6. Conclusions

In this article, the DLE distribution has been derived using the survival function from the continuous OPLE distribution. An extensive study of this distribution is conducted, including mathematical characteristic functions such as the CDF, PMF, SF, and HRF, the shape of the PMF, and the failure rate (HRF). Additionally, the calculation of statistical properties has been investigated, such as mean, moments, DI, skewness, kurtosis, and CV. The proposed distribution has been suitable for modeling over-dispersed count data. The model parameter has been estimated using classical (ML) and various Bayesian estimation methods. The Bayesian estimators have been computed using three loss functions, including SELF, LINEX, and GELF, under informative and non-informative priors. Comprehensive simulation results have been carried out to compare classical and Bayesian methods. Since the conditional posteriors of the parameters cannot be obtained in any standard forms, using MCMC to draw samples from the joint posterior of the parameter has been suggested. Based on the simulation study, the best estimator obtained was the Bayesian method under GELF with positive weight (q = +1.5). Furthermore, according to the RMSEs of all estimation methods, we found that the estimators perform well and approach the values of the estimated parameters as the sample size increases. The performance of the proposed distribution has been illustrated empirically using physical and medical applications. Some statistical criteria were used to show that this distribution has been the best fit for these datasets, including AIC, BIC, K-S test, etc. The DLE distribution has shown superiority over the other competitive discrete distributions. The results demonstrated that the DLE distribution was will-fitting to all datasets, particularly the physical dataset. The applications also have been employed to compare the classical and Bayesian methods. For future work, censored data can be considered, and the regression model can be studied for this distribution.