On the Discretization of the Weibull-G Family of Distributions: Properties, Parameter Estimates, and Applications of a New Discrete Distribution

Abstract

In this article, we introduce a new three-parameter distribution called the discrete Weibull exponential (DWE) distribution, based on the use of a discretization technique for the Weibull-G family of distributions. This distribution is noteworthy, as its probability mass function presents both symmetric and asymmetric shapes. In addition, its related hazard function is tractable, exhibiting a wide range of shapes, including increasing, increasing–constant, uniform, monotonically increasing, and reversed J-shaped. We also discuss some of the properties of the proposed distribution, such as the moments, moment-generating function, dispersion index, Rényi entropy, and order statistics. The maximum likelihood method is employed to estimate the model’s unknown parameters, and these estimates are evaluated through simulation studies. Additionally, the effectiveness of the model is examined by applying it to three real data sets. The results demonstrate that, in comparison to the other considered distributions, the proposed distribution provides a better fit to the data.

1. Introduction

The use of statistical distributions is crucial in fitting various types of data. Traditional distributions do not always capture the diverse shapes and behaviors found in natural data, leading to the development of new families of distributions to accommodate more complex data. On the other hand, discrete distributions are essential for measuring quantities (e.g., the lifetime of products) on a non-continuous scale, prompting researchers to introduce new discrete distributions. These distributions, however, have limitations in modeling complex data, encouraging the development of more flexible options. As a result, interest has increased in discretizing existing continuous models to achieve a better fit for a larger range of data sets. A statistical method relying on creating new discrete G classes of probability models from existing continuous classes has recently been derived. For instance, Ref. [1] presented and defined the discrete Gompertz-G family of distributions following [2]. A new G family of continuous distributions, named the exponential generalized-G family, was presented in [3]. Following [4], Ref. [5] introduced and discussed the discrete Rayleigh G family of distributions, whereas [6] suggested a discrete analogue of the Weibull-G family.

Recently, Ref. [7] presented a new method, called transformed-transformer, to generate new families of distributions. They chose the weight function $W\left(G\right(x,\delta \left)\right)=-log(1-G(x,\delta \left)\right)$, where $G(x,\delta )$ is a cumulative distribution function of any baseline distribution with parameter vector $\delta$, in order to introduce the Weibull-G, beta-exponential-G, and gamma-G families of distributions.

In this paper, $W\left(G\right(x,\delta \left)\right)=-log(1-G(x,\delta \left)\right)$, which has been discussed in [7,8,9], is specifically considered. The cumulative distribution function (CDF) of the Weibull-G distribution is defined as

$$\begin{array}{cc}\hfill F(x,\alpha ,\beta ,\delta )={\int }_0^{-log(1-G(x,\delta \left)\right)}{\displaystyle \frac{\alpha }{\beta }}{\left({\displaystyle \frac{t}{\beta }}\right)}^{\alpha -1}{e}^{-{\left({\displaystyle \frac{t}{\beta }}\right)}^{\alpha }}dt& =1-e^{-{\left({\displaystyle \frac{-log(1-G(x,\delta \left)\right)}{\beta }}\right)}^{\alpha }},\hfill \end{array}$$

(1)

with corresponding probability density function (PDF):

$$f(x;\alpha ,\beta ,\delta )={\displaystyle \frac{\alpha }{\beta }}\left({\displaystyle \frac{g(x,\delta )}{1-G(x,\delta )}}\right){\left({\displaystyle \frac{-log(1-G(x,\delta \left)\right)}{\beta }}\right)}^{\alpha -1}{e}^{-{\left({\displaystyle \frac{-log(1-G(x,\delta \left)\right)}{\beta }}\right)}^{\alpha }}.$$

(2)

The survival function (SF) and the hazard rate function (HRF) are, respectively, given by

$$S(x;\alpha ,\beta ,\delta )=e^{-{\left({\displaystyle \frac{-log(1-G(x,\delta \left)\right)}{\beta }}\right)}^{\alpha }},$$

(3)

$$h(x;\alpha ,\beta ,\delta )={\displaystyle \frac{\alpha }{\beta }}\left({\displaystyle \frac{g(x,\delta )}{1-G(x,\delta )}}\right){\left({\displaystyle \frac{-log(1-G(x,\delta \left)\right)}{\beta }}\right)}^{\alpha -1}.$$

(4)

Over the last few decades, several discretized forms of continuous distributions have been derived to model different discrete data sets. The most notable discretization approach is the survival discretization method proposed by [10]. It is a novel technique that utilizes the survival function of any continuous distribution to develop a new discrete distribution. The probability mass function of a discrete distribution has been defined in [10] as follows:

$$P(X=x)=S\left(x\right)-S(x+1),x=0,1,2,\dots ,$$

(5)

where $S\left(x\right)$ is the survival function of the continuous distribution, defined as

$$S\left(x\right)=P(X\ge x)=1-F(x;\mathsf{\Theta }),$$

(6)

where $F(x;\mathsf{\Theta })$ is a CDF of the continuous distribution and $\mathsf{\Theta }$ is a vector of parameters.

This approach has recently been extended to define some new discrete families of probability distributions. For example, Ref. [1] introduced the discrete Gompertz-G family of distributions and studied the discrete Gompertz exponential, discrete Gompertz–Weibull, and discrete Gompertz–inverse Weibull distributions; Ref. [5] studied the discrete Rayleigh family of distributions and proposed the discrete Rayleigh–Weibull distribution; Ref. [11] presented the odd Perks-G family of distributions and presented the discrete odd Perks exponential distribution; Ref. [12] introduced a discrete exponential generalized-G family of distributions and introduced the discrete exponential generalized Weibull distribution; and Ref. [13] proposed a discrete analogue of the odd Weibull-G family of distributions and studied the discrete odd Weibull–Geometric and the discrete odd Weibull–Inverse Weibull distributions.

Therefore, the main focus of this paper is to utilize the discretization technique proposed in [10] to construct a new discrete distribution that is more flexible in fitting data. To illustrate this, in this study, we use the survival discretization technique to convert the continuous Weibull-G (W-G) family to a discrete family, named the discrete Weibull-G (DW-G) family, in order to develop a three-parameter distribution, which is called the discrete Weibull exponential (DWE) distribution.

Starting with the continuous Weibull-G family proposed in [7], and utilizing the discretization approach from [10], the CDF of THE discrete Weibull-G (DW-G) family can be formulated as follows:

$$F(x;\alpha ,\beta ,\delta )=1-e^{-{\left({\displaystyle \frac{-log(1-G(x+1,\delta \left)\right)}{\beta }}\right)}^{\alpha }}.$$

(7)

and the corresponding SF of the DW-G family can be derived as follows:

$$S(x;\alpha ,\beta ,\delta )=e^{-{\left({\displaystyle \frac{-log(1-G(x+1,\delta \left)\right)}{\beta }}\right)}^{\alpha }}.$$

(8)

Therefore, the probability mass function (PMF) of the DW-G family can be expressed as follows:

$$f(x;\alpha ,\beta ,\delta )=e^{-{\left({\displaystyle \frac{-log(1-G(x,\delta \left)\right)}{\beta }}\right)}^{\alpha }}-e^{-{\left({\displaystyle \frac{-log(1-G(x+1,\delta \left)\right)}{\beta }}\right)}^{\alpha }}.$$

(9)

and based on Equations (8) and (9), the HRF can then be written as follows:

$$h(x;\alpha ,\beta ,\delta )=1-{\displaystyle \frac{e^{-{\left(\frac{-log(1-G(x+1,\delta \left)\right)}{\beta }\right)}^{\alpha }}}{e^{-{\left(\frac{-log(1-G(x,\delta \left)\right)}{\beta }\right)}^{\alpha }}}}.$$

(10)

The remainder of this article is organized as follows: Section 2 introduces the DWE distribution as well as its survival and hazard rate functions. Section 3 explores some fundamental statistical properties of the DWE distribution. Section 4 discusses the estimation of its distribution parameters using the method of maximum likelihood estimation (MLE). Section 5 supplies a simulation study in order to evaluate the estimates. Section 6 provides three real data applications which exemplify the adequacy of the proposed distribution. Finally, Section 7 concludes this paper.

2. Discrete Weibull Exponential Distribution (DWE)

If X is the exponential random variable with parameter $\theta >0$, then its CDF is given by

$$G\left(x\right)=1-e^{-\theta x},x>0,\theta >0.$$

(11)

Substituting $G\left(x\right)$ in Equations (7) and (9), the DWE distribution has the following CDF and PMF, respectively:

$$f(x;\alpha ,\beta ,\theta )=e^{-{\left({\displaystyle \frac{\theta x}{\beta }}\right)}^{\alpha }}-e^{-{\left({\displaystyle \frac{\theta (x+1)}{\beta }}\right)}^{\alpha }},$$

(12)

$$F(x;\alpha ,\beta ,\theta )=1-e^{-{\left({\displaystyle \frac{\theta (x+1)}{\beta }}\right)}^{\alpha }}.$$

(13)

Moreover, the SF and HRF are, respectively, given as

$$S\left(x\right)=e^{-{\left({\displaystyle \frac{\theta x}{\beta }}\right)}^{\alpha }},$$

(14)

$$h\left(x\right)=1-{\displaystyle \frac{e^{-{\left(\frac{\theta (x+1)}{\beta }\right)}^{\alpha }}}{e^{-{\left(\frac{\theta x}{\beta }\right)}^{\alpha }}}}.$$

(15)

Some PMF and HRF plots for the DWE model generated under selected parameter values are shown in Figure 1 and Figure 2. From Figure 1, it can be seen that the PMF plots of the DWE distribution exhibit various shapes, which include symmetric, right skewed, increasing, decreasing, J-shaped, and reversed J-shaped curves. Additionally, Figure 2 reveals that the HRF of the DWE distribution presents a variety of shapes. In particular, it can be increasing, increasing–constant, decreasing, uniform, or monotonically increasing. These observations strongly suggest the great flexibility of the DWE distribution in fitting data.

3. Statistical Properties of the DWE Distribution

Some properties of the DWE distribution are detailed in this section, including the quantile function, moments, moment-generating function, dispersion index, coefficient of variation, Rényi entropy, and order statistics.

3.1. The Quantile Function and the Median

The $q^{th}$ quantile of DWE distribution can be obtained by

$$x_q=⌈{\displaystyle \frac{\beta }{\theta }}{\left\{log\left({\displaystyle \frac{1}{1-q}}\right)\right\}}^{{\displaystyle \frac{1}{\alpha }}}-1⌉,$$

(16)

where $\lceil .\rceil$ is the ceiling function, which returns the smallest integer greater than or equal to its argument.

In particular, if we set q = 0.5, we obtain the value of the median of the proposed distribution; in particular, the median M is given as:

$$M=⌈{\displaystyle \frac{\beta }{\theta }}\{log(2){\}}^{{\displaystyle \frac{1}{\alpha }}}-1⌉.$$

(17)

3.2. Moments

The rth moment of a random variable X, which follows the DWE distribution, can be calculated as

$$\grave{{\mu }_r}=E\left(x^r\right)=\sum _{x=0}^{\infty }{x}^{r}f\left(x\right)=\sum _{x=1}^{\infty }[x^r-{(x-1)}^r]e^{-{\left({\displaystyle \frac{\theta x}{\beta }}\right)}^{\alpha }},$$

where $f\left(x\right)$ is the PMF of DWE distribution.

Therefore,

$$\grave{{\mu }_1}=E\left(x\right)=\sum _{x=1}^{\infty }{e}^{-{\left({\displaystyle \frac{\theta x}{\beta }}\right)}^{\alpha }},$$

$$\grave{{\mu }_2}=E\left(x^2\right)=\sum _{x=1}^{\infty }(2x-1)e^{-{\left({\displaystyle \frac{\theta x}{\beta }}\right)}^{\alpha }},$$

$$\grave{{\mu }_3}=E\left(x^3\right)=\sum _{x=1}^{\infty }(3x^2-3x+1)e^{-{\left({\displaystyle \frac{\theta x}{\beta }}\right)}^{\alpha }},$$

$$\grave{{\mu }_4}=E\left(x^4\right)=\sum _{x=1}^{\infty }(4x^3-6x^2+4x-1)e^{-{\left({\displaystyle \frac{\theta x}{\beta }}\right)}^{\alpha }}.$$

Thus, the variance of the DWE distribution is obtained as follows:

$${\sigma }^2=\grave{{\mu }_2}-{\grave{{\mu }_1}}^2=\sum _{x=1}^{\infty }(2x-1)e^{-{\left({\displaystyle \frac{\theta x}{\beta }}\right)}^{\alpha }}-{\left[\sum _{x=1}^{\infty }{e}^{-{\left({\displaystyle \frac{\theta x}{\beta }}\right)}^{\alpha }}\right]}^2.$$

Moreover, the skewness and kurtosis can be found, respectively, as follows:

$$skewness\left(sk\right)={\displaystyle \frac{\grave{{\mu }_3}-3\grave{{\mu }_2}\grave{{\mu }_1}+2{\grave{{\mu }_1}}^2}{{{\sigma }^2}^{\frac{3}{2}}}},$$

$$kurtosis\left(K\right)={\displaystyle \frac{\grave{{\mu }_4}-4\grave{{\mu }_3}\grave{{\mu }_1}+6\grave{{\mu }_2}{\grave{{\mu }_1}}^2-3{\grave{{\mu }_1}}^4}{{\left({\sigma }^2\right)}^2}}.$$

3.3. The Moment-Generating Function

Assume that X is a non-negative random variable (RV), which follows the DWE distribution. Then, the moment-generating function of the RV X can be derived as follows:

$$\begin{gathered} M_x\left(t\right)=E\left(e^{tx}\right)={\sum }_{x=0}^{\infty }{e}^{tx}p\left(x\right) \\ ={\sum }_{x=0}^{\infty }{e}^{tx}[e^{-{\left({\displaystyle \frac{\theta x}{\beta }}\right)}^{\alpha }}-e^{-{\left({\displaystyle \frac{\theta (x+1)}{\beta }}\right)}^{\alpha }}]=1+{\sum }_{x=1}^{\infty }[e^{tx}-e^{t(x-1)}]e^{-{\left({\displaystyle \frac{\theta x}{\beta }}\right)}^{\alpha }}. \end{gathered}$$

3.4. The Dispersion Index and Coefficient of Variation

The dispersion index (DSI) of the DWE distribution is defined as

$$DSI={\displaystyle \frac{{\sum }_{x=1}^{\infty }(2x-1)e^{-{\left(\frac{\theta x}{\beta }\right)}^{\alpha }}-{\left[{\sum }_{x=1}^{\infty }{e}^{-{\left(\frac{\theta x}{\beta }\right)}^{\alpha }}\right]}^2}{{\sum }_{x=1}^{\infty }{e}^{-{\left(\frac{\theta x}{\beta }\right)}^{\alpha }}}}.$$

Furthermore, the coefficient of variation (COV) of the DWE distribution is calculated as follows

$$cov\left(x\right)={\displaystyle \frac{var{\left(x\right)}^{\frac{1}{2}}}{mean}}={\displaystyle \frac{{\left\{{\sum }_{x=1}^{\infty }(2x-1)e^{-{\left(\frac{\theta x}{\beta }\right)}^{\alpha }}-{\left[{\sum }_{x=1}^{\infty }{e}^{-{\left(\frac{\theta x}{\beta }\right)}^{\alpha }}\right]}^2\right\}}^{0.5}}{{\sum }_{x=1}^{\infty }{e}^{-{\left(\frac{\theta x}{\beta }\right)}^{\alpha }}}}.$$

Some numerical results for the mean, variance, skewness (sk), kurtosis (K), DSI, and COV are presented in

Table 1. Numerical results for mean, variance (var), skewness (sk), kurtosis (K), dispersion index (DSI), and coefficient of variation (COV).

$\mathit{\theta }$	$\mathit{\beta }$	$\mathit{\alpha }$	Mean	var	Sk	K	DSI	COV
0.9	0.8	0.8	0.63613	1.4820365	2.623497	11.67308	2.178932	1.764302
1.0	0.8	0.8	0.53651	1.1562765	2.692359	12.16114	1.955228	1.625485
1.1	0.8	0.8	0.46758	0.9618487	2.755160	12.46055	1.986928	1.709372
1.2	0.8	0.8	0.40572	0.7907006	2.827562	12.82022	2.118363	2.330601
1.3	0.8	0.8	0.35036	0.6422556	2.868490	13.08421	1.194890	2.651182
1.4	0.8	0.8	0.31317	0.5464306	2.915547	13.27730	1.565657	2.502524
1.5	0.8	0.8	0.27463	0.4599740	3.007003	13.82224	1.392622	2.460666
1.6	0.8	0.8	0.24485	0.3937704	3.050039	13.98168	1.111485	2.028944
1.7	0.8	0.8	0.22099	0.3426884	3.105343	14.22292	2.144781	3.195817
0.7	0.9	2.5	0.63839	0.3318948	0.2150545	2.273680	0.6397306	1.0325782
0.7	1.0	2.5	0.76310	0.3838390	0.2229947	2.591419	0.4885216	0.7450759
0.7	1.1	2.5	0.89339	0.4471207	0.2431885	2.708928	0.4970760	0.7233519
0.7	1.2	2.5	1.01903	0.5134084	0.2575146	2.758525	0.4499450	0.6674504
0.7	1.3	2.5	1.14110	0.5835564	0.2629662	2.781759	0.3957219	0.6228665
0.7	1.4	2.5	1.27459	0.6669239	0.2675351	2.782106	0.5070707	0.6369117
0.7	1.5	2.5	1.39779	0.7482641	0.2737800	2.796096	0.5004979	0.5936862
0.7	1.6	2.5	1.52671	0.8363944	0.2810906	2.799882	0.4797980	0.5476073
0.7	1.7	2.5	1.65514	0.9356285	0.2849057	2.799589	0.6695992	0.6572667
0.5	0.5	0.9	0.65577	1.2787146	2.273438	9.417977	2.0405504	1.719686
0.5	0.5	1.0	0.58157	0.9162312	2.025603	7.960426	1.5180573	1.442058
0.5	0.5	1.1	0.53670	0.7277665	1.831957	6.841325	1.4842629	1.466665
0.5	0.5	1.2	0.49981	0.5927157	1.657637	5.929515	1.2475738	1.564041
0.5	0.5	1.3	0.46878	0.4969840	1.498745	5.175420	0.9437229	1.835876
0.5	0.5	1.4	0.45180	0.4368885	1.351888	4.502136	0.8915989	1.474663
0.5	0.5	1.5	0.43161	0.3865797	1.242426	4.034442	0.7800224	1.316580
0.5	0.5	1.6	0.41877	0.3507332	1.133300	3.592376	0.7650417	1.289626
0.5	0.5	1.7	0.40692	0.3244679	1.049193	3.242691	0.8106648	1.373050

.

From Table 1, The following is clear: As $\theta$ increases under fixed values of $\alpha$ and $\beta$, the mean and variance decrease, whereas the sk and K increase. As $\beta$ increases under fixed values of $\theta$ and $\alpha$, the mean, variance, sk, and K increase. As $\alpha$ increases under fixed values of $\theta$ and $\beta$, the mean, variance, sk, and K decrease. In addition, the DSI can be less or greater than 1, which means that it is suitable for representing under-or over-dispersed phenomena.

3.5. The Rényi Entropy

The Rényi Entropy (RE), proposed in [14], is a fundamental entropy function. It is obtained as follows:

$${\displaystyle {\gamma }_R\left(\rho \right)=\frac{1}{1-\rho }log\left(\sum _{x=0}^{\infty }{f}^{\rho }\left(x\right)\right),}$$

(18)

where $\rho >0$ and $\rho \ne 0$.

Substituting Equation (12) into the entropy function, we obtain

$${\displaystyle {\gamma }_R\left(\rho \right)=\frac{1}{1-\rho }log\left(\sum _{x=0}^{\infty }{\left[e^{-{\left(\frac{\theta x}{\beta }\right)}^{\alpha }}-e^{-{\left(\frac{\theta (x+1)}{\beta }\right)}^{\alpha }}\right]}^{\rho }\right).}$$

3.6. The Order Statistic

Assume that $x_1,x_2,...,x_n$ are randomly sampled from the DWE distribution and let $X_{r:n}$ denote the rth order statistic.

Consider $F_r(x;\alpha ,\beta ,\theta )$; the CDF of the ith order statistic for a random sample $x_1,x_2,\dots ,x_n$ from the DWE distribution is given by

$$F_{r:n}=\sum _{i=r}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right){\left(F\left(x\right)\right)}^i{[1-F\left(x\right)]}^{n-i}.$$

(19)

using the binomial expansion equation,

$${(1-x)}^{n-i}=\sum _{j=0}^{n-i}{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{n-i}{j}}\right)x^j,$$

(20)

and, substituting Equation (13) into Equation (19), the CDF of the order statistic can be presented as

$$\begin{array}{cc}{F}_{r:n}\hfill & =\sum _{i=r}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right){\left(F\left(x\right)\right)}^i{[1-F\left(x\right)]}^{n-i}\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & =\sum _{i=r}^n\sum _{j=0}^{n-i}{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n-i}{j}}\right){\left(F\left(x\right)\right)}^i{\left(F\left(x\right)\right)}^j\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill & =\sum _{i=r}^n\sum _{j=0}^{n-i}{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n-i}{j}}\right){\left(F\left(x\right)\right)}^{i+j}\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill & =\sum _{i=r}^n\sum _{j=0}^{n-i}{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n-i}{j}}\right){(1-e^{-{\left({\displaystyle \frac{\theta (x+1)}{\beta }}\right)}^{\alpha }})}^{i+j}.\hfill \end{array}$$

(24)

The corresponding PMF of the rth order statistic can be obtained by applying the formula discussed in [15]:

$$\begin{array}{c}\hfill f_r\left(z\right)=F_r\left(z\right)-F_r(z-1).\end{array}$$

(25)

Thus,

$$f_r(x;\alpha ,\beta ,\theta )=\sum _{i=r}^n\sum _{j=0}^{n-i}{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n-i}{j}}\right)\left[{(1-e^{-{\left({\displaystyle \frac{\theta (x+1)}{\beta }}\right)}^{\alpha }})}^{i+j}-{(1-e^{-{\left({\displaystyle \frac{\theta x}{\beta }}\right)}^{\alpha }})}^{i+j}\right].$$

(26)

In particular, setting $r=1$ and $r=n$ in Equation (26), we can obtain the PMFs of the minimum and maximum order statistics, respectively.

4. Parameter Estimation for DWE Distribution

The MLE approach was used to estimate the parameters for the DWE distribution. The steps are as follows: First, define the log-likelihood function. Second, consider the three parameters $\theta ,\beta ,$ and $\alpha$, and differentiate the log-likelihood function with respect to each parameter. Finally, set the derivative equations to zeros. The likelihood function $L(x;\theta ,\beta ,\alpha )$ for the DWE distribution is given as follows

$$L(X;\theta ,\beta ,\alpha )=\prod _{i=1}^{n}F\left(X_i\right)=\prod _{i=1}^n\left[e^{-{\left({\displaystyle \frac{\theta x}{\beta }}\right)}^{\alpha }}-e^{-{\left({\displaystyle \frac{\theta (x+1)}{\beta }}\right)}^{\alpha }}\right].$$

Then, the log-likelihood function for the DWE distribution can be defined as

$$\mathsf{\ell }(\theta ,\beta ,\alpha ;\mathbf{x})=\sum _{i=1}^{n}log\left[e^{-{\left({\displaystyle \frac{\theta x}{\beta }}\right)}^{\alpha }}-e^{-{\left({\displaystyle \frac{\theta (x+1)}{\beta }}\right)}^{\alpha }}\right].$$

(27)

The derivatives of Equation (27), with respect to the DWE distribution parameters, are obtained as follows:

$$\begin{array}{ccc}& {\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }\theta }}\mathsf{\ell }(\theta ,\beta ,\alpha ;\mathbf{x})=\hfill & \hfill {\displaystyle \frac{\alpha }{{\beta }^{\alpha }}}{\theta }^{\alpha -1}\sum _{i=1}^n{\displaystyle \frac{{(x_i+1)}^{\alpha }{\mathrm{e}}^{-{\left\{\theta \left(x_i+1\right)/\beta \right\}}^{\alpha }}-x_i^{\alpha }{\mathrm{e}}^{-{\left(\theta x_i/\beta \right)}^{\alpha }}}{{\mathrm{e}}^{-{\left(\theta x_i/\beta \right)}^{\alpha }}-{\mathrm{e}}^{-{\left\{\theta \left(x_i+1\right)/\beta \right\}}^{\alpha }}}},\end{array}$$

(28)

$$\begin{array}{ccc}& {\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }\beta }}\mathsf{\ell }(\theta ,\beta ,\alpha ;\mathbf{x})=\hfill & \hfill {\displaystyle \frac{\alpha }{{\beta }^{\alpha +1}}}{\theta }^{\alpha }\sum _{i=1}^n{\displaystyle \frac{x_i^{\alpha }{\mathrm{e}}^{-{\left(\theta x_i/\beta \right)}^{\alpha }}-{(x_i+1)}^{\alpha }{\mathrm{e}}^{-{\left(\theta (x_i+1)/\beta \right)}^{\alpha }}}{{\mathrm{e}}^{-{\left(\theta x_i/\beta \right)}^{\alpha }}-{\mathrm{e}}^{-{\left\{\theta \left(x_i+1\right)/\beta \right\}}^{\alpha }}}},\end{array}$$

(29)

$$\begin{array}{cc}& {\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }\alpha }}\mathsf{\ell }(\theta ,\beta ,\alpha ;\mathbf{x})=\hfill \\ & {\left({\displaystyle \frac{\theta }{\beta }}\right)}^{\alpha }\sum _{i=1}^n{\displaystyle \frac{{(x_i+1)}^{\alpha }ln\left\{\theta \left(x_i+1\right)/\beta \right\}{\mathrm{e}}^{-{\left\{\theta \left(x_i+1\right)/\beta \right\}}^{\alpha }}-x_i^{\alpha }ln\left(\theta x_i/\beta \right){\mathrm{e}}^{-{\left(\theta x_i/\beta \right)}^{\alpha }}}{{\mathrm{e}}^{-{\left(\theta x_i/\beta \right)}^{\alpha }}-{\mathrm{e}}^{-{\left\{\theta \left(x_i+1\right)/\beta \right\}}^{\alpha }}}}.\hfill \end{array}$$

(30)

Hence, the MLEs of the parameters $\theta ,\beta ,$ and $\alpha$ can be obtained by equating Equations (28)–(30) to zero, and solving them either analytically or numerically, using a method such as the Newton–Raphson iteration method. Moreover, using an R function like optim and nim, the estimators can automatically be obtained by maximizing Equation (27).

5. Simulation Study

In this section, three cases are presented to examine the performance of the MLEs of the DWE distribution parameters. Some values were considered for the parameters, as follows:

Case I:

$\theta =0.5,\beta =0.6,\alpha =1.1$.

Case II:

$\theta =0.2,\beta =0.5,\alpha =2.4$.

Case III:

$\theta =0.05,\beta =0.4,\alpha =1.3$.

The mean square error (MSE) was applied to evaluate the MLE, $\widehat{\omega }$ for every parameter. The number of iterations for each case in the simulation was nsim = 10,000. The mean square error (MSE) can be calculated as

$$MSE\left(\widehat{\omega }\right)={\displaystyle \frac{{\sum }_{i=1}^{nsim}{(\widehat{{\omega }_i}-{\omega }_{tr})}^2}{nsim}}.$$

The Monte Carlo simulation method was used. The MLEs of the parameters and their corresponding MSEs are listed in

Table 2. Simulation results for the DWE MLEs and MSEs with different sample sizes for the three considered cases.

Sample Size	Parameter	Case I		Case II		Case III
		MLE	MSE	MLE	MSE	MLE	MSE
$n=30$	$\theta$	0.5074528	0.003882414	0.2025147	2.214930 × 10−4	0.05138392	5.946656 × 10−5
	$\beta$	0.5964033	0.002817521	0.5015774	3.806611 × 10−5	0.40195144	6.627253 × 10−6
	$\alpha$	1.1929766	0.087645668	2.5298629	1.904817 × 10−1	1.35938594	4.455467 × 10−2
$n=100$	$\theta$	0.5025115	0.0010940215	0.2011377	6.642254 × 10−5	0.05058812	1.721367 × 10−5
	$\beta$	0.5989836	0.0007626512	0.5013801	1.236618 × 10−5	0.40171244	3.818087 × 10−6
	$\alpha$	1.1241868	0.0179010381	2.4408913	4.721245 × 10−2	1.31816245	1.151682 × 10−2
$n=200$	$\theta$	0.5010536	0.0005443831	0.2007682	3.185856 × 10−5	0.05029812	8.172484 × 10−6
	$\beta$	0.5996454	0.0003819548	0.5014658	7.656253 × 10−6	0.40149683	2.769827 × 10−6
	$\alpha$	1.1114204	0.0085155657	2.4174074	2.289356 × 10−2	1.30845216	5.572183 × 10−3
$n=500$	$\theta$	0.5006724	0.0002112696	0.2006683	1.355192 × 10−5	0.05026356	3.346773 × 10−6
	$\beta$	0.5998296	0.0001473779	0.5012536	3.362149 × 10−6	0.40140377	2.338581 × 10−6
	$\alpha$	1.1045133	0.0033042406	2.4075469	8.797369 × 10−3	1.30385604	2.169917 × 10−3

.

From Figure 3, Figure 4 and Figure 5, and Table 2, the MSE clearly becomes smaller as the sample size increases, and the MLEs become nearer to the true parameters.

6. Application

In this section, the flexibility of the DWE distribution is illustrated by fitting three real data sets using six different distributions, in addition to the proposed DWE distribution. The PMFs for the six distributions used in the comparison are as follows:

(1)

Geometric Distribution (Geom)

$f\left(x\right)={(1-\theta )}^{x-1}\theta ;0<\theta <1.$

(2)

Poisson distribution (Pois)

$f\left(x\right)={\displaystyle \frac{{\lambda }^{x}exp(-\lambda )}{x!}},\lambda >0,x=0,1,2,..$

(3)

Discrete Weibull–Geometric Distribution (DWGeom) [16]:

$f\left(x\right)={\displaystyle \frac{(1-p)[{\rho }^{x^{\alpha }}-{\rho }^{{(x+1)}^{\alpha }}]}{(1-p{\rho }^{x^{\alpha }})[1-p{\rho }^{{(x+1)}^{\alpha }}]}};0<p<1,0<\rho <1,\alpha >0.$

(4)

Discrete fréchet (dfréchet) [17]:

$f\left(x\right)=exp\left[-{\left({\displaystyle \frac{\beta }{x+1}}\right)}^{\alpha }\right]-exp\left[{\left({\displaystyle \frac{\beta }{x}}\right)}^{\alpha }\right];\alpha ,\beta >0.$

(5)

Discrete extended odd Weibull exponential distribution (DEOWE) [18]:

$f\left(x\right)={\left(1+\beta {\left[e^{\lambda x}-1\right]}^{\alpha }\right)}^{-{\displaystyle \frac{1}{\beta }}}-{\left(1+\beta {\left[e^{\lambda (x+1)}-1\right]}^{\alpha }\right)}^{-{\displaystyle \frac{1}{\beta }}};\alpha ,\beta ,\lambda >0.$

(6)

Discrete Weibull Distribution (DWD) [19]:

$f\left(x\right)=q^{y^{\beta }}-q^{{(y+1)}^{\beta }};y=0,1,2,..,\beta >0,q\in (0,1).$

The parameters of fitted distributions were estimated using the maximum likelihood method (i.e., maximizing the log-likelihood). Goodness-of-fit (GOF) criteria, including the Corrected Akaike Information Criterion (AICc) [20], Akaike’s Information Criterion (AIC) [21], Bayesian Information Criterion (BIC) [22], and Hannan–Quinn Information Criterion (HQIC) [23], were used to discriminate the best model. Plots were generated for comparison between the DWE distribution and the other distributions, and the Kolmogorov–Simornov (K-S) test was conducted to determine the p-value for each distribution. The formulas for the GOF criteria are listed below:

$$AIC=-2\mathsf{\ell }+2h,$$

(31)

$$AICc=AIC+{\displaystyle \frac{2h(h+1)}{n-h-1}},$$

(32)

$$BIC=-2\mathsf{\ell }+h\ast log\left(n\right),$$

(33)

$$HQIC=-2\mathsf{\ell }+2h\ast log\left(log\right(n\left)\right),$$

(34)

where ℓ is the log-likelihood function evaluated for the MLEs, h is the number of estimated parameters for the model, and n is the sample size.

6.1. First Data Set

The first data set is presented in [24]. It contains 26 observations that indicate the failure times for a specific product. This information has also been used in [11,25]. Figure 6 and Figure 7 show the p-p plot of the fitted distributions and the empirical CDFs for this data set. A summary of the MLE results for the parameters, along with the SEs, log-likelihood, AICc, AIC, BIC, and HQIC for each distribution is provided in

Table 3. MLEs (with their corresponding SEs in parentheses ) and GOF criteria for the first data set.

Distributions	DWE	GEOM	Pois	DWGeom	DFrechet	DEOWE	DW
Parameter estimation	$\widehat{\theta }$ = 0.0505	$\widehat{\theta }$ = 0.0233	$\widehat{\lambda }$ = 42.884	$\widehat{p}$ = 0.1166	$\widehat{\beta }$ = 35.906	$\widehat{\alpha }$ = 2.6158	$\widehat{q}$ = 0.9941
	(0.0995)	(0.0045)	(1.2843)	(0.3936)	(2.1594)	(0.2543)	(0.0037)
	$\widehat{\beta }$ = 2.4137	$---$	$---$	$\widehat{\rho }$ = 0.9959	$\widehat{\alpha }$ = 3.4671	$\widehat{\beta }$ = 0.6271	$\widehat{\beta }$ = 1.3578
	(4.751)	$---$	$---$	(0.0021)	(0.5085)	(0.8295)	(0.1654)
	$\widehat{\alpha }$ = 4.3467	$---$	$---$	$\widehat{\alpha }$ = 1.4229	$---$	$\widehat{\lambda }$ = 0.0162	$---$
	(0.6987)	$---$	$---$	(0.1404)	$---$	(0.0006)	$---$
−logL	100.1117	123.4158	114.7385	116.5324	104.1828	103.9409	116.8963
AICc	207.3144	248.9983	231.6436	240.1557	212.8872	214.9728	238.3143
AIC	206.2235	248.8316	231.4769	239.0648	212.3655	213.8819	237.7925
BIC	209.9978	250.0897	232.7350	242.8391	214.8817	217.6562	240.3087
HQIC	207.3103	249.1939	231.8392	240.1517	213.0901	214.9688	238.5171
p-Value	6.7108 × 10−1	1.2006 × 10−4	3.9658 × 10−2	3.6631 × 10−3	4.0413 × 10−1	5.5648 × 10−1	1.4349 × 10−3

.

6.2. Second Data Set

The second data set was reported in [26] and provides 61-day COVID-19 data recorded between 13 June and 12 August of 2021 in Italy. This data set includes newly reported cases on a daily basis. Figure 8 and Figure 9 show the p-p plots of the fitted distributions and the empirical CDFs for the second data set, respectively. A summary of the results of the MLEs of the parameters along with the SEs, the log-likelihood, AICc, AIC, BIC, and HQIC for each distribution is provided in

Table 4. MLEs (with their corresponding SEs in parentheses ) and GOF criteria for the second data set.

Distributions	DWE	GEOM	Pois	DWGeom	DFrechet	DEOWE	DW
Parameter estimation	$\widehat{\theta }$ = 0.0362	$\widehat{\theta }$ = 0.0442	$\widehat{\lambda }$ = 22.623	$\widehat{p}$ = 0.0211	$\widehat{\beta }$ = 14.604	$\widehat{\alpha }$ = 7.8043	$\widehat{q}$ = 0.9925
	(0.0836 )	(0.0055)	(0.609)	(0.36163 )	(1.2822)	(1.5493)	(0.0031)
	$\widehat{\beta }$ = 0.9462	$---$	$---$	$\widehat{\rho }$ = 0.9923	$\widehat{\alpha }$ = 1.5544	$\widehat{\beta }$ = 17.578	$\widehat{\beta }$ = 1.5259
	(2.1857 )	$---$	$---$	(0.0044)	(0.1407)	(6.7639)	(0.1236)
	$\widehat{\alpha }$ = 1.9434	$---$	$---$	$\widehat{\alpha }$ = 1.5108	$---$	$\widehat{\lambda }$ = 0.0883	$---$
	(0.1868)	$---$	$---$	(0.1346)	$---$	(0.012)	$---$
−logL	235.5084	249.8884	349.2768	238.6766	245.6704	248.2538	238.3603
AICc	477.4378	501.8445	700.6214	483.7743	495.5477	502.9286	480.9276
AIC	477.0168	501.7767	700.5536	483.3533	495.3408	502.5076	480.7207
BIC	483.3494	503.8876	702.6645	489.6859	499.5625	508.8402	484.9424
HQIC	479.4986	502.6040	701.3809	485.8351	496.9953	504.9894	482.3752
p-Value	8.3373 × 10−1	1.0404 × 10−3	3.245 × 10−4	4.3862 × 10−1	2.4088 × 10−1	6.9468 × 10−3	2.0653 × 10−1

.

6.3. Third Data Set

The third data set was reported in [27], and provides 101 observations. It includes the fatigue life (at 18 cycles per second) of 6061-T6 aluminum coupons that were cut parallel to the direction of rolling and oscillated. Figure 10 and Figure 11 show the p-p plots of the fitted distributions and the empirical CDFs for the third data set, respectively. A summary of the results of the MLEs of the parameters along with the SEs, the log-likelihood, AICc, AIC, BIC, and HQIC for each distribution is provided in

Table 5. MLEs (with their corresponding SEs in parentheses) and GOF criteria for the third dataset.

Distributions	DWE	GEOM	Pois	DWGeom	DFrechet	DEOWE	DW
Parameter estimation	$\widehat{\theta }$ = 0.0165	$\widehat{\theta }$ = 0.0146	$\widehat{\lambda }$ = 68.34	$\widehat{p}$ = 0.165	$\widehat{\beta }$ = 50.9829	$\widehat{\alpha }$ = 5.4788	$\widehat{q}$ = 0.9952
	(0.0188)	(0.0014)	(0.8267)	(0.1901)	(3.7091)	(0.5196)	(0.0013)
	$\widehat{\beta }$ = 1.2586	$---$	$---$	$\widehat{\rho }$ = 0.9952	$\widehat{\alpha }$ = 1.4642	$\widehat{\beta }$ = 11.866	$\widehat{\beta }$ = 1.2614
	(1.4371)	$---$	$---$	(0.0014)	(0.0834)	(1.8745)	(0.0663)
	$\widehat{\alpha }$ = 3.2033	$---$	$---$	$\widehat{\alpha }$ = 1.2306	$---$	$\widehat{\lambda }$ = 0.0163	$---$
	(0.232921)	$---$	$---$	(0.070508)	$---$	(0.001101)	$---$
−logL	454.2379	521.7143	678.2322	509.1632	513.9630	511.3819	503.4051
AICc	914.7257	1045.4694	1358.5051	1024.5765	1032.0497	1029.0138	1010.9339
AIC	914.4757	1045.4286	1358.4643	1024.3265	1031.9260	1028.7638	1010.8102
BIC	922.2912	1048.0338	1361.0695	1032.1420	1037.1363	1036.5793	1016.0205
HQIC	917.6388	1046.4829	1359.5187	1027.4895	1034.0347	1031.9269	1012.9189
P-Value	6.5356 × 10−1	3.0345 × 10−12	6.5006 × 10−5	1.1408 × 10−9	1.895 × 10−6	2.9103 × 10−12	1.5569 × 10−9

.

From Table 3, Table 4 and Table 5, it is evident that the DWE distribution outperformed the other distributions when considering the AICc, AIC, BIC, and HQIC. In addition, Figure 6, Figure 8 and Figure 10, as well as Figure 7, Figure 9 and Figure 11 support this conclusion.

7. Conclusions

In this work, discretization of a continuous family of distributions was studied. This discretized family of distributions is called the discrete Weibull-G family of distributions, and the three-parameter DWE distribution is presented as a member of this family. Both the PMF and HRF of the DWE distribution have shapes that appear to be promising for capturing the various behaviors of data. Furthermore, some of the statistical properties of this distribution were studied. Its three parameters were estimated using the maximum likelihood method. Three cases with different sample sizes and different DWE parameter values were used to assess the performance of the MLEs of the DWE distribution parameters. Three real data sets were utilized to examine the efficiency of the DWE distribution, in comparison with various commonly used distributions. The presented figures demonstrate the usefulness of the proposed distribution. The obtained results indicate that the proposed DWE distribution represents a highly flexible distribution with good potential for modeling real data.