# Fuzzy Reliability Analysis of the COVID-19 Mortality Rate Using a New Modified Kies Kumaraswamy Model

Authors: Fathy H. Riad; Bader Alruwaili; Ehab M. Almetwally; Eslam Hussam
Journal: Journal of Mathematics
 (2022)

---

## Abstract

In this paper, we developed a novel superior distribution, demonstrated and derived its mathematical features, and assessed its fuzzy reliability function. The novel distribution has numerous advantages, including the fact that its CDf and PDf have a closed shape, making it particularly relevant in many domains of data science. We used both conventional and Bayesian approaches to make various sorts of estimations. A simulation research was carried out to investigate the performance of the classical and Bayesian estimators. Finally, we fitted a COVID-19 mortality real data set to the suggested distribution in order to compare its efficiency to that of its rivals.


---

## Body

## 1. Introduction

Nowadays we face a huge waves of viruses such as COVID-19, which attracted a great attention in the last two years. However, as statisticians we must play an important role in understanding and modelling the COVID-19 infections, so we had to make a statistical model that is capable of fitting and modelling them, whether it is continuous or discrete random variable. In this paper, we made a lot of effort to choose an appropriate model which is superior in fitting the COVID-19 infections in many countries. At last, we settled to introduce a new superior distribution as mixture of other ones to overcome the defects of the baseline distribution.Over the years and decades, modelling for lifetime distributions has garnered considerable attention, and their popularity has increased as the relevance of modelling phenomena and pandemics has grown. Researchers in distribution theory choose to model the data whether by adding a new parameter that increases the flexibility of the distribution of interest or by creating a new distribution family. Modelling is extremely useful in a wide variety of sectors, including commerce, economics, dependability, and health research. See [1] for more reading. In the next paragraph we will discuss in details the new proposed distribution.We developed a statistical model for COVID-19 mortality data in Saudi Arabia in this study; for further information about the other models, refer to Kumar [2], Khakharia et al. [3], Shafiq et al. [4], Sindhu et al. [5], Wang [6], Lalmuanawma et al. [7], and Bullock et al. [8].The new proposed model is a combination between the Kumaraswamy distribution’s (Ku) and the modified Kies distribution. The CDF of Ku distribution and PDF with parametersα,β>0 were put forth by [9] in the following manner:(1)Gx;β=1−1−xαβ,0<x<1,α,β>0,(2)gx;β=αβxα−11−xαβ−1,0<x<1,α,β>0.Many authors worked on extensions of the Kies distribution as an example; see Kumar and Dharmaja [10] who worked on the exponentiated Kies distribution. Also, Dey et al. [11] worked on the modified Kies distribution when the sample they worked on censored data specially under type II progressive censored sample. Al-Babtain et al. [12] developed a different distribution family based on the modified Kies (MK) family. Let us say that Gz;δ is the baseline CDF depending on a parameter vector δ; (3) defines the CDF of the MK family. Kumar and Dharmaja examined the characteristics of the exponentiated Kies distribution as we can see in reference [10]. One of the authors who have created a MK distribution for a type II progressive censored sample; in addition, the estimation of the distribution parameters has been obtained (see Dey et al. [11] for more information). Referring to Al-Babtainet al. [12], if Gz;δ is the baseline CDF δ, then Gz;δ is the MK family’s CDF that may be expressed as follows:(3)Fx;Θ=1−e−Gx;Δ/1−Gx;Δθ,θ>0,where Θ is the parameter vector θ,Δ and Δ is parameter vector of Gx.The PDF of (3) is as follows:(4)fx;Θ=θgx;ΔGx;Δθ−11−Gx;Δθ+1e−Gx;Δ/1−Gx;δθ,θ>0,Using the distributions above, the two-parameter modified Kies Kumaraswamy (MKKu) is produced in this paper. The MKKu has a number of interesting properties. The suggested MKKu distribution has an extremely adaptable PDF since it may be positive skewed, negative skewed, or symmetric, which allows for more flexibility for tail. It is capable of simulating declining, increasing, bathtub, and reverse-J hazard rates. Also, the suggested distribution has a precise closed-form CDF and is extremely easy to manage. These advantages make the distribution an attractive choice for applications in a variety of fields, including biological life testing, dependability, and actuarial data. This article showed one real-world data application and demonstrated via modelling that the novel distribution is a great rival to many well-known and conventional distributions with the parameters of the scale and shape such as type II power Topp-Leone inverse exponential (Bantan et al. [13]), Topp-Leone generalized exponential (Kunjiratanachot et al. [14]), Kumaraswamy, beta, Gompertz Lomax (Oguntunde et al. [15]), alpha power inverted TL (Ibrahim et al. [16]), and MK exponential (Al-Babtain et al. [12]) distributions.The paper is arranged and organized as follows. In Section2, the MKKu distribution is obtained. In Section 3, some mathematical characteristics of the MKKu distribution are discussed. In Section 4, we establish an MKKu distribution estimate technique. In Section 5, a fuzzy reliability function was obtained. In Section 6, we present the simulation study for the estimation techniques and concluded our major findings in this section. In Section 7, we acquired applications of real-world data analysis. Section 8 summarises and concludes the article. Future research will include the development of an extension for the bivariate MK inverted Topp-Leone based on copula as in [17]. We propose to investigate a unique application of the MKITL distribution using a censored sample; see [18] for further information.


## 2. MKKu Distribution

By employing the Kumaraswamy distribution with these parametersα,β and the CDF specified (for 0 < x < 1), (1) and (2) are used. We construct the CDF for MKKu distribution by putting the Ku distribution’s CDF into (3).(5)Fx;Θ=1−e−1−xα−β−1θ,α,β,θ>0,The corresponding PDF is(6)fx;Θ=θαβxα−11−1−xαβθ−11−xαβθ+1e−1−xα−β−1θ,α,β,θ>0,where Θ is parameter vector θ,α,β.The survival function of the MKKu distribution is represented by(7)Sx;Θ=e−1−xα−β−1θ.The function of hazard rate (HR) of the MKKu distribution is given by(8)hrz;Θ=θαβxα−11−1−xαβθ−11−xαβθ+1.Figures1 and 2 provide specific plots of the MKKu distribution for the values indicated for θ,α and β in the equation of the MKKu distribution. It can be seen by the diagrams provided in Figure 2 that the MKKu distribution HR function may be increased, lowered, and shaped like a bathtub. In contrast to a Ku distribution, which is a bad model for the data and phenomena that indicate growing, decreasing forms, and failure rates of the bathtub, the MKKu distribution is more adaptable in terms of evaluating lifespan data than its rival; this is one of its advantages.Figure 1
The PDF of the MKKu distribution for certain parameter values.Figure 2
HR of the MKKu distribution for different parameter values.


## 3. The Suggested Distribution Statistical Properties

### 3.1. A General Expansion of the MKKu Density

In this article, we provide a linear representation for the MK family and utilise it to construct a suitable linear representation for the MKKu distribution.(9)fx;Θ=θ∑j=0∞−1jj!αβθxα−11−xα−βθj+1−11−1−xαβθj+1−1.The last equation of the MKKu distribution may be reformulated using the PDF and CDF of the Ku distribution.(10)fx;Θ=2βαθ∑j,k=0∞−1j+kj!θj+1−1kxα−11−xαβk−θj+1−1.Equation (10) denotes the Ku density with parameter α,βk−θj+1.


### 3.2. Quantile for the MKKu Distribution

In this part of the paper, we drive the formula of the MKKu distribution’s quantile function; for example, let us say thatx=F−1x,ΘQ is derived by inverting (5) as follows:(11)x=1−1+−ln1−U1/θ−1/β1/α,0<U<1.The first quartile, denoted by the symbolU1, the second quartile, denoted by the symbol U2, and the third quartile, denoted by the symbol U3, are produced by putting U = 0.25, 05, 0.75, accordingly in (11).


### 3.3. The Proposed Distribution Related Moments

Therth of the distribution can be find by using (10) as(12)μr´=Exr=βθ∑j,k=0∞−1j+kj!θj+1−1k∫01αxα+r−11−xαβk−θj+1−1dx=βθ∑j,k=0∞−1j+kj!θj+1−1kBetar+αα,βk−θj+1.


## 3.1. A General Expansion of the MKKu Density

In this article, we provide a linear representation for the MK family and utilise it to construct a suitable linear representation for the MKKu distribution.(9)fx;Θ=θ∑j=0∞−1jj!αβθxα−11−xα−βθj+1−11−1−xαβθj+1−1.The last equation of the MKKu distribution may be reformulated using the PDF and CDF of the Ku distribution.(10)fx;Θ=2βαθ∑j,k=0∞−1j+kj!θj+1−1kxα−11−xαβk−θj+1−1.Equation (10) denotes the Ku density with parameter α,βk−θj+1.


## 3.2. Quantile for the MKKu Distribution

In this part of the paper, we drive the formula of the MKKu distribution’s quantile function; for example, let us say thatx=F−1x,ΘQ is derived by inverting (5) as follows:(11)x=1−1+−ln1−U1/θ−1/β1/α,0<U<1.The first quartile, denoted by the symbolU1, the second quartile, denoted by the symbol U2, and the third quartile, denoted by the symbol U3, are produced by putting U = 0.25, 05, 0.75, accordingly in (11).


## 3.3. The Proposed Distribution Related Moments

Therth of the distribution can be find by using (10) as(12)μr´=Exr=βθ∑j,k=0∞−1j+kj!θj+1−1k∫01αxα+r−11−xαβk−θj+1−1dx=βθ∑j,k=0∞−1j+kj!θj+1−1kBetar+αα,βk−θj+1.


## 4. Estimation Techniques for Distribution Parameters

Using Bayesian and non-Bayesian estimate approaches, we are able to solve the estimation issue for the MKKu distribution parameters. The maximum likelihood estimators (MLE), maximum product of spacings, Anderson–Darling, Cramér–von-Misse, and Bayesian approaches based on the squared error loss function (SELF) are used. These non-Bayesian methods have been discussed to estimate parameters for different models as per [19–21], while for Bayesian estimation, see [22–24].


### 4.1. The MLE Estimator

Supposen is a random sample of any size, then likelihood function of the proposed distribution can be written as follows:(13)LΘ=θnαnβne−∑i=1n1−xiα−β−1θ∏i=1nxiα−11−1−xiαβθ−11−xiαβθ+1.For such MKKu distribution, the log-likelihood formula is represented by(14)lΘ=nlnθ+lnβ+lnα−∑i=1n1−xiα−β−1θ+α−1∑i=1nlnxi+θ−1∑i=1nln1−1−xiαβ−βθ+1∑i=1nln1−xiα.We will differentiate with respect to every parameter in the distribution(15)∂lΘ∂θ=nθ−∑i=1n1−xiα−β−1θln1−xiα−β−1+∑i=1nln1−1−xiαβ−β∑i=1nln1−xiα,∂lΘ∂α=nα−θβ∑i=1nxiαlnxi1−xiα−β−11−xiα−β−1θ−1+∑i=1nlnxi+θ−1β∑i=1nxiαlnxi1−xiαβ−11−1−xiαβ+βθ+1∑i=1nxiαlnxi1−xiα,∂lΘ∂β=nβ+θ∑i=1n1−xiα−β1−xiα−β−1θ−1ln1−xiα−θ−1∑i=1n1−xiαβln1−xiα1−1−xiαβ−β∑i=1nln1−xiα.We can evaluate the estimates of the three parametersθ, α, and β by determining the greatest value for (14) and then by computing its first derivative in terms of θ, α, and β. Using the Newton-Rapshon approach, the R program may be used to maximize the log-likelihood and yield the MLE.


### 4.2. Maximum Product of Spacings (MPS) Method

MPS approach was established by Cheng and Amin [25] and has been employed in censored applications as [26–28]. Assume x1<⋯<xn. Given a random sample of size n, the MKKu distribution’s uniform spacing may be written as(16)DiΘ=Fxi,Θ−Fxi−1,Θ,i=1,…,n+1,DiΘ indicate the uniform spacings, Fx0,Θ=0, Fxn+1,Θ=1 and ∑i=1n+1DiΘ=1. The MPS estimators (MPSE) of the MKKu parameters can be found by maximizing(17)GΘ=1n+1∑i=1n+1loge−1−xi−1α−β−1θ−e−1−xiα−β−1θ,with respect to θ,α, and β.The MPSE of the MKKu parameters can found by solving(18)∂GΘ∂θ=1n+1∑i=1n+1e−1−xiα−β−1θ1−xiα−β−1θlog1−xiα−β−1e−1−xi−1α−β−1θ−e−1−xiα−β−1θ−e−1−xi−1α−β−1θ1−xi−1α−β−1θlog1−xi−1α−β−1e−1−xi−1α−β−1θ−e−1−xiα−β−1θ,∂GΘ∂α=θβn+1∑i=1n+1e−1−xiα−β−1θ1−xi−1α−β−1θ−11−xi−1α−β−1xiαlogxie−1−xi−1α−β−1θ−e−1−xiα−β−1θ−e−1−xi−1α−β−1θ1−xi−1α−β−1θ−11−xi−1α−β−1xi−1αlogxi−1e−1−xi−1α−β−1θ−e−1−xiα−β−1θ,∂GΘ∂β=θn+1∑i=1n+1e−1−xi−1α−β−1θ1−xi−1α−β−1θ−11−xi−1α−βlog1−xi−1αe−1−xi−1α−β−1θ−e−1−xiα−β−1θ−e−1−xiα−β−1θ1−xiα−β−1θ−11−xiα−βlog1−xiαe−1−xi−1α−β−1θ−e−1−xiα−β−1θ.


### 4.3. Cramer–von-Mises (CVME) Method

This method may be calculated using the difference between both the CDF and empirical distribution function estimations by Luceno [29]. The CVME values for the MKKu parameters are produced by minimizing the next function in terms of θ,α, and β:(19)CΘ=112+∑i=1n1−e−1−xiα−β−1θ−2i−12n2.Further, the CVME follow by solving the next nonlinear equations(20)∂CΘ∂θ=2∑i=1n1−e−1−xiα−β−1θ−2i−12ne−1−xiα−β−1θ1−xiα−β−1θlog1−xiα−β−1,(21)∂CΘ∂α=2∑i=1n1−e−1−xiα−β−1θ−2i−12ne−1−xiα−β−1θθβxiαlogxi1−xiα−β−1θ−11−xiα−β−1,(22)∂CΘ∂β=−2θ∑i=1n1−e−1−xiα−β−1θ−2i−12ne−1−xiα−β−1θ1−xiα−β−1θ−11−xiα−βlog1−xiα,Solve these equations to get the estimates (20)–(22).


### 4.4. Anderson–Darling Method

Another sort of minimal distance estimators is the (ADE) Anderson–Darling estimators, minimizing the ADE of the MKKu parameters with respect toθ,α, and β. Additionally, the ADE may be produced by solving nonlinear equations, which yields the ADE of the MKKu parameters estimates.(23)AΘ=−n−1n∑i=1n2i−1log1−e−1−xiα−β−1θ−1−xn+1−iα−β−1θ2.


### 4.5. Bayesian Estimation Method

This section contains Bayesian estimations ofθ,α, and β. We assume that θ,α, and β have gamma priors; also, the gamma prior density was first used. The following are the gamma priors for distribution parameters:(24)ℂ1θ∝θφ1−1e−d1θ,θ>0,φ1,d1>0,ℂ2α∝αφ2−1e−d2α,α>0,φ2,d2>0,ℂ3β∝βφ3−1e−d3β,β>0,φ3,d3>0.As we know the two priors are independent of each other, so the joint prior ofθ and β is obtained as(25)ℂθ,α,β∝θφ1−1αφ2−1βφ3−1exp−θd1+αd2+βd3,θ,α,β>0.To get acceptable and superior values for the independent joint prior’s hyperparameters, we may employ the MLE method’s estimate and variance-covariance matrix. The calculated hyperparameters may be represented by equating the mean and variance of gamma priors:(26)φ1=1/B∑l=1Bθ^i21/B−1∑l=1Bθ^i−1/B∑l=1Bθ^i2,φ2=1/B∑l=1Bα^i21/B−1∑l=1Bα^i−1/B∑l=1Bα^i2,φ3=1/B∑l=1Bβ^i21/B−1∑l=1Bβ^i−1/B∑l=1Bβ^i2,d1=1/B∑l=1Bθ^i1/B−1∑l=1Bθ^i−1/B∑l=1Bθ^i2,d2=1/B∑l=1Bα^i1/B−1∑l=1Bα^i−1/B∑l=1Bα^i2,d3=1/B∑l=1Bβ^i1/B−1∑l=1Bβ^i−1/B∑l=1Bβ^i2,where B is the number of iteration.By multiplying (13) by (25) and making some simplifications, the posterior distribution ℂ∗θ,β,α is formed as follows:(27)ℂ∗Θ∝θn+φ1−1αn+φ2−1βn+φ3−1e−θd1+αd2+βd3e−∑i=1n1−xiα−β−1θ∏i=1nxiα−11−1−xiαβθ−11−xiαβθ+1,AsuΘ=uθ,β,α, the Bayesian estimations of the distribution parameter using the squared error loss function (SELF) is(28)u˜SEΘ=EuΘ.It is obvious that SELF estimations ofuθ,β,α in (28) is considered as the division of more than one integration over each other. As the multiple integrals are hard to deal with, so we used Mathematica 11 to apply an approximation technique, which is very useful in solving these kinds of integration. So in this context, we used the Markov Chain Monte Carlo (MCMC) technique to get an approximate value of integrals. The important methods of the MCMC technique is the Metropolis-Hastings (MH) algorithm. The following approach is used to produce random samples from conditional posterior densities of the MKKu distribution:(29)ℂ∗θ|α,β,x∝θn+φ1−1e−θd1e−∑i=1n1−xiα−β−1θ∏i=1n1−1−xiαβθ−11−xiαβθ+1,ℂ∗α|θ,β,x∝αn+φ2−1e−αd2e−∑i=1n1−xiα−β−1θ∏i=1nxiα−11−1−xiαβθ−11−xiαβθ+1,ℂ∗β|θ,α,x∝βn+φ3−1e−βd3e−∑i=1n1−xiα−β−1θ∏i=1n1−1−xiαβθ−11−xiαβθ+1,


## 4.1. The MLE Estimator

Supposen is a random sample of any size, then likelihood function of the proposed distribution can be written as follows:(13)LΘ=θnαnβne−∑i=1n1−xiα−β−1θ∏i=1nxiα−11−1−xiαβθ−11−xiαβθ+1.For such MKKu distribution, the log-likelihood formula is represented by(14)lΘ=nlnθ+lnβ+lnα−∑i=1n1−xiα−β−1θ+α−1∑i=1nlnxi+θ−1∑i=1nln1−1−xiαβ−βθ+1∑i=1nln1−xiα.We will differentiate with respect to every parameter in the distribution(15)∂lΘ∂θ=nθ−∑i=1n1−xiα−β−1θln1−xiα−β−1+∑i=1nln1−1−xiαβ−β∑i=1nln1−xiα,∂lΘ∂α=nα−θβ∑i=1nxiαlnxi1−xiα−β−11−xiα−β−1θ−1+∑i=1nlnxi+θ−1β∑i=1nxiαlnxi1−xiαβ−11−1−xiαβ+βθ+1∑i=1nxiαlnxi1−xiα,∂lΘ∂β=nβ+θ∑i=1n1−xiα−β1−xiα−β−1θ−1ln1−xiα−θ−1∑i=1n1−xiαβln1−xiα1−1−xiαβ−β∑i=1nln1−xiα.We can evaluate the estimates of the three parametersθ, α, and β by determining the greatest value for (14) and then by computing its first derivative in terms of θ, α, and β. Using the Newton-Rapshon approach, the R program may be used to maximize the log-likelihood and yield the MLE.


## 4.2. Maximum Product of Spacings (MPS) Method

MPS approach was established by Cheng and Amin [25] and has been employed in censored applications as [26–28]. Assume x1<⋯<xn. Given a random sample of size n, the MKKu distribution’s uniform spacing may be written as(16)DiΘ=Fxi,Θ−Fxi−1,Θ,i=1,…,n+1,DiΘ indicate the uniform spacings, Fx0,Θ=0, Fxn+1,Θ=1 and ∑i=1n+1DiΘ=1. The MPS estimators (MPSE) of the MKKu parameters can be found by maximizing(17)GΘ=1n+1∑i=1n+1loge−1−xi−1α−β−1θ−e−1−xiα−β−1θ,with respect to θ,α, and β.The MPSE of the MKKu parameters can found by solving(18)∂GΘ∂θ=1n+1∑i=1n+1e−1−xiα−β−1θ1−xiα−β−1θlog1−xiα−β−1e−1−xi−1α−β−1θ−e−1−xiα−β−1θ−e−1−xi−1α−β−1θ1−xi−1α−β−1θlog1−xi−1α−β−1e−1−xi−1α−β−1θ−e−1−xiα−β−1θ,∂GΘ∂α=θβn+1∑i=1n+1e−1−xiα−β−1θ1−xi−1α−β−1θ−11−xi−1α−β−1xiαlogxie−1−xi−1α−β−1θ−e−1−xiα−β−1θ−e−1−xi−1α−β−1θ1−xi−1α−β−1θ−11−xi−1α−β−1xi−1αlogxi−1e−1−xi−1α−β−1θ−e−1−xiα−β−1θ,∂GΘ∂β=θn+1∑i=1n+1e−1−xi−1α−β−1θ1−xi−1α−β−1θ−11−xi−1α−βlog1−xi−1αe−1−xi−1α−β−1θ−e−1−xiα−β−1θ−e−1−xiα−β−1θ1−xiα−β−1θ−11−xiα−βlog1−xiαe−1−xi−1α−β−1θ−e−1−xiα−β−1θ.


## 4.3. Cramer–von-Mises (CVME) Method

This method may be calculated using the difference between both the CDF and empirical distribution function estimations by Luceno [29]. The CVME values for the MKKu parameters are produced by minimizing the next function in terms of θ,α, and β:(19)CΘ=112+∑i=1n1−e−1−xiα−β−1θ−2i−12n2.Further, the CVME follow by solving the next nonlinear equations(20)∂CΘ∂θ=2∑i=1n1−e−1−xiα−β−1θ−2i−12ne−1−xiα−β−1θ1−xiα−β−1θlog1−xiα−β−1,(21)∂CΘ∂α=2∑i=1n1−e−1−xiα−β−1θ−2i−12ne−1−xiα−β−1θθβxiαlogxi1−xiα−β−1θ−11−xiα−β−1,(22)∂CΘ∂β=−2θ∑i=1n1−e−1−xiα−β−1θ−2i−12ne−1−xiα−β−1θ1−xiα−β−1θ−11−xiα−βlog1−xiα,Solve these equations to get the estimates (20)–(22).


## 4.4. Anderson–Darling Method

Another sort of minimal distance estimators is the (ADE) Anderson–Darling estimators, minimizing the ADE of the MKKu parameters with respect toθ,α, and β. Additionally, the ADE may be produced by solving nonlinear equations, which yields the ADE of the MKKu parameters estimates.(23)AΘ=−n−1n∑i=1n2i−1log1−e−1−xiα−β−1θ−1−xn+1−iα−β−1θ2.


## 4.5. Bayesian Estimation Method

This section contains Bayesian estimations ofθ,α, and β. We assume that θ,α, and β have gamma priors; also, the gamma prior density was first used. The following are the gamma priors for distribution parameters:(24)ℂ1θ∝θφ1−1e−d1θ,θ>0,φ1,d1>0,ℂ2α∝αφ2−1e−d2α,α>0,φ2,d2>0,ℂ3β∝βφ3−1e−d3β,β>0,φ3,d3>0.As we know the two priors are independent of each other, so the joint prior ofθ and β is obtained as(25)ℂθ,α,β∝θφ1−1αφ2−1βφ3−1exp−θd1+αd2+βd3,θ,α,β>0.To get acceptable and superior values for the independent joint prior’s hyperparameters, we may employ the MLE method’s estimate and variance-covariance matrix. The calculated hyperparameters may be represented by equating the mean and variance of gamma priors:(26)φ1=1/B∑l=1Bθ^i21/B−1∑l=1Bθ^i−1/B∑l=1Bθ^i2,φ2=1/B∑l=1Bα^i21/B−1∑l=1Bα^i−1/B∑l=1Bα^i2,φ3=1/B∑l=1Bβ^i21/B−1∑l=1Bβ^i−1/B∑l=1Bβ^i2,d1=1/B∑l=1Bθ^i1/B−1∑l=1Bθ^i−1/B∑l=1Bθ^i2,d2=1/B∑l=1Bα^i1/B−1∑l=1Bα^i−1/B∑l=1Bα^i2,d3=1/B∑l=1Bβ^i1/B−1∑l=1Bβ^i−1/B∑l=1Bβ^i2,where B is the number of iteration.By multiplying (13) by (25) and making some simplifications, the posterior distribution ℂ∗θ,β,α is formed as follows:(27)ℂ∗Θ∝θn+φ1−1αn+φ2−1βn+φ3−1e−θd1+αd2+βd3e−∑i=1n1−xiα−β−1θ∏i=1nxiα−11−1−xiαβθ−11−xiαβθ+1,AsuΘ=uθ,β,α, the Bayesian estimations of the distribution parameter using the squared error loss function (SELF) is(28)u˜SEΘ=EuΘ.It is obvious that SELF estimations ofuθ,β,α in (28) is considered as the division of more than one integration over each other. As the multiple integrals are hard to deal with, so we used Mathematica 11 to apply an approximation technique, which is very useful in solving these kinds of integration. So in this context, we used the Markov Chain Monte Carlo (MCMC) technique to get an approximate value of integrals. The important methods of the MCMC technique is the Metropolis-Hastings (MH) algorithm. The following approach is used to produce random samples from conditional posterior densities of the MKKu distribution:(29)ℂ∗θ|α,β,x∝θn+φ1−1e−θd1e−∑i=1n1−xiα−β−1θ∏i=1n1−1−xiαβθ−11−xiαβθ+1,ℂ∗α|θ,β,x∝αn+φ2−1e−αd2e−∑i=1n1−xiα−β−1θ∏i=1nxiα−11−1−xiαβθ−11−xiαβθ+1,ℂ∗β|θ,α,x∝βn+φ3−1e−βd3e−∑i=1n1−xiα−β−1θ∏i=1n1−1−xiαβθ−11−xiαβθ+1,


## 5. Fuzzy Reliability

The concepts of dependability and HR function are probabilities that describe the life time (a random variable) from the start of a failure to the point where we need to adjust the manufacturing process. The use of dependability and HR functions has been broadened now that fuzzy factors have been included to these functions. The lifespan probability of the system’s components is treated as accurate numbers in classic reliability models. In the real world, however, this accuracy of system lifetimes is not true since the values of system parameters acquired via experiment, faulty measurement, human judgement, or guess are all subject to some uncertainty [30]. As a result, incorporating the idea of fuzziness will be more acceptable. As a result, while studying system behaviour or addressing system reliability, it will be more suitable to include the idea of fuzziness. The membership functions are utilised to describe the level of fuzziness associated with the lifespan data or system parameters in order to generate the fuzzy parameters [31]. The fuzzy set theory may be used to explain the real world in a realistic and practical way. Zadeh [32] was the one who introduced it.


### 5.1. Basic Concepts in the Theory of Fuzzy Sets

The data type utilised to estimate the parameters, as well as the trustworthiness of the probability distribution, is critical to the correctness of the findings we acquire. As a result, the data type has to be given. Fuzzy data are one of these sorts of data, and it is one of the new and essential developments in statistics since many occurrences in our actual world have no precise limits. They, too, lack precision in their measurements. As a result, we will go over some fundamental notions in the theory of fuzzy sets [31].LetT be a continuous random variable that represents a system’s failure time (component). Chen and Pham [33] may then calculate the fuzzy dependability using the fuzzy probability formula:(30)R˜t=PT>t=∫t∞μxfxdx,0≤t≤x<∞,where μx is a membership function that indicates the degree to which each element of a given universe belongs to the fuzzy set.Now, assume thatμx is(31)μx=0,x≤t1,x−t1t2−t1,t1<x<t2,t1≥0,1,x≥t2.The lifespanxγ for μx can be calculated using the computational approach of the function of fuzzy numbers and corresponds to a given value of γ−Cut,γ∈0,1,as follows (Chen and Pham [33]):μx=γ⟶x−t1/t2−t1=γ, then(32)xγ≤t1,γ=0,xγ=t1+γt2−t1,0<γ<1,xγ≥t2,γ=1.Thus, the fuzzy reliability values can be obtained for all values ofγ, and we consider fuzzy reliability of MKKu distribution depends on the definition of fuzzy reliability as per the following equations:Ifγ=0,(33)R˜tγ=0=∫t1t1fxdx=0.Ifγ has value,(34)R˜t0<γ<1=∫t1xγμxfxdx=∫t1xγμxθαβxα−11−1−xαβθ−11−xαβθ+1e−1−xα−β−1θdx,where xγ=t1+γt2−t1, and if γ=1,(35)R˜tγ=1=∫t1t2fxdx=∫t1t2θαβxα−11−1−xαβθ−11−xαβθ+1e−1−xα−β−1θdx=e−1−t1α−β−1θ−e−1−T2α−β−1θ.


## 5.1. Basic Concepts in the Theory of Fuzzy Sets

The data type utilised to estimate the parameters, as well as the trustworthiness of the probability distribution, is critical to the correctness of the findings we acquire. As a result, the data type has to be given. Fuzzy data are one of these sorts of data, and it is one of the new and essential developments in statistics since many occurrences in our actual world have no precise limits. They, too, lack precision in their measurements. As a result, we will go over some fundamental notions in the theory of fuzzy sets [31].LetT be a continuous random variable that represents a system’s failure time (component). Chen and Pham [33] may then calculate the fuzzy dependability using the fuzzy probability formula:(30)R˜t=PT>t=∫t∞μxfxdx,0≤t≤x<∞,where μx is a membership function that indicates the degree to which each element of a given universe belongs to the fuzzy set.Now, assume thatμx is(31)μx=0,x≤t1,x−t1t2−t1,t1<x<t2,t1≥0,1,x≥t2.The lifespanxγ for μx can be calculated using the computational approach of the function of fuzzy numbers and corresponds to a given value of γ−Cut,γ∈0,1,as follows (Chen and Pham [33]):μx=γ⟶x−t1/t2−t1=γ, then(32)xγ≤t1,γ=0,xγ=t1+γt2−t1,0<γ<1,xγ≥t2,γ=1.Thus, the fuzzy reliability values can be obtained for all values ofγ, and we consider fuzzy reliability of MKKu distribution depends on the definition of fuzzy reliability as per the following equations:Ifγ=0,(33)R˜tγ=0=∫t1t1fxdx=0.Ifγ has value,(34)R˜t0<γ<1=∫t1xγμxfxdx=∫t1xγμxθαβxα−11−1−xαβθ−11−xαβθ+1e−1−xα−β−1θdx,where xγ=t1+γt2−t1, and if γ=1,(35)R˜tγ=1=∫t1t2fxdx=∫t1t2θαβxα−11−1−xαβθ−11−xαβθ+1e−1−xα−β−1θdx=e−1−t1α−β−1θ−e−1−T2α−β−1θ.


## 6. Simulation Results

In this part, we estimated the MKKu parameters using two different approaches and compared their performance using a simulation study. We explore several sample sizesn=30,80,150 for various parameter choices of θ,α, and β. We produce random samples of N=1000 from the MKKu distribution. We derive the relative bias (RB) values of the estimates, the mean square error (MSE), and the accompanying fuzzy reliability for each estimate as follows:(36)Rf1=R˜tγ=0.5,Rf2=R˜tγ=0.7,Rf3=R˜tγ=0.9.The effectiveness of different estimators is quantified in terms of RB, MSE, and confidence interval, with the most effective methods having MSE values near to zero. The R software is used to acquire the simulation results. Tables1 and 2 show the estimates RB, MSE, and confidence interval for MLE and Bayesian.Table 1
RB and MSE of the MLE, MPS, CVM, AD, and Bayesian estimate for MKKu distribution forα=3,β=0.7.
α=3,β=0.7MLEMPSCVMADBayesianθRBMSERBMSERBMSERBMSERBMSE330θ0.04520.1505−0.03190.11500.01250.09420.00140.0883−0.00010.0250α0.02740.0635−0.01930.04150.00770.02930.00340.0263−0.00240.0216β0.02320.0025−0.01850.00170.00670.00170.00190.0013−0.00230.0016Rf1−0.96390.0009−0.96040.0008−0.96220.0008−0.96190.0007−0.96160.0004Rf2−0.92750.0022−0.92400.0017−0.92530.0020−0.92520.0018−0.92430.0012Rf3−0.42280.0036−0.42400.0028−0.41830.0039−0.42080.0033−0.41470.002880θ0.01630.0564−0.02040.04130.00110.01340.00130.0134−0.00060.0155α0.01160.0409−0.01250.01340.00070.00380.00080.0039−0.00270.0140β0.00940.0013−0.01160.00050.00040.00030.00020.0003−0.00310.0007Rf1−0.96270.0003−0.96090.0003−0.96190.0002−0.96190.0002−0.96180.0002Rf2−0.92570.0008−0.92390.0007−0.92460.0006−0.92470.0005−0.92470.0005Rf3−0.41750.0012−0.41740.0010−0.41490.0012−0.41520.0011−0.41580.0010150θ0.00560.0274−0.01790.0237−0.00940.0186−0.00550.0147−0.00090.0077α0.00050.0175−0.01080.0073−0.00370.0058−0.00300.00450.00140.0073β0.00320.0006−0.00710.0003−0.00110.0004−0.00030.00020.00260.0004Rf1−0.96170.0002−0.96060.0002−0.96110.0002−0.96130.0001−0.96160.0001Rf2−0.92430.0004−0.92320.0004−0.92360.0004−0.92380.0004−0.92420.0003Rf3−0.41100.0007−0.41110.0006−0.41080.0007−0.41060.0006−0.41130.00060.530θ0.05710.01960.02760.01680.06440.01350.04930.01210.02120.0066α0.04590.5967−0.05330.4173−0.00950.0766−0.02480.1783−0.00220.0249β0.10030.0658−0.02670.03870.05760.05050.02140.0395−0.00910.0155Rf1−0.83260.0001−0.84590.0001−0.83240.0002−0.83740.0001−0.83860.0001Rf2−0.95940.0002−0.96270.0002−0.95930.0004−0.96060.0002−0.96090.0002Rf3−0.76600.0005−0.78520.0004−0.76580.0008−0.77290.0005−0.77480.000480θ0.03840.01550.00890.00500.01870.00520.01840.00510.00700.0030α0.03870.9714−0.03470.16970.00650.0661−0.00590.11680.00070.0194β0.05590.0538−0.03540.01340.02390.02000.00370.0150−0.00250.0077Rf1−0.83693.484E − 05−0.84300.0000−0.83660.0001−0.83830.0000−0.83863.091E − 05Rf2−0.96057.956E − 05−0.96200.0001−0.96040.0001−0.96080.0001−0.96097.884E − 05Rf3−0.77261.544E − 04−0.78140.0002−0.77220.0002−0.77460.0002−0.77521.427E − 04150θ0.01800.00490.00290.00250.02070.00310.01940.00270.00440.0015α0.00560.2288−0.02320.0877−0.01070.0529−0.01900.0501−0.00170.0080β0.02050.0145−0.01630.00790.00720.0095−0.00490.00740.00540.0040Rf1−0.83691.899E − 05−0.84091.820E − 05−0.83673.140E − 05−0.83772.241E − 05−0.83781.795E − 05Rf2−0.96054.338E − 05−0.96154.305E − 05−0.96047.036E − 05−0.96075.058E − 05−0.96074.304E − 05Rf3−0.77288.393E − 05−0.77868.364E − 05−0.77251.333E − 04−0.77409.664E − 05−0.77428.309E − 05Table 2
RB and MSE of the MLE, MPS, CVM, AD, and Bayesian estimate for MKKu distribution forθ=3,β=0.7.
θ=3,β=0.7MLEMPSCVMADBayesianαRBMSERBMSERBMSERBMSERBMSE0.530θ0.10560.07020.23610.09340.13280.02850.11480.04300.01680.0082α0.17550.0944−0.08890.0821−0.04190.02480.01150.05100.00700.0135β0.23930.1891−0.04630.11910.04510.06350.09270.1105−0.00570.0152Rf1−0.90785.716E − 05−0.91020.0000−0.91260.0001−0.91030.0001−0.91203.454E − 05Rf2−0.86901.197E − 04−0.87170.0001−0.87640.0002−0.87260.0001−0.87477.104E − 05Rf3−0.87792.157E − 04−0.87960.0002−0.88510.0004−0.88110.0002−0.88281.261E − 0480θ0.02010.03270.14210.04120.05440.01470.05180.01860.02300.0048α0.16680.0732−0.07290.04720.00400.02180.01350.02760.00010.0077β0.20030.1465−0.04790.07090.04440.04790.04690.0517−0.01240.0098Rf1−0.90861.574E − 05−0.90930.0000−0.91000.0000−0.90970.0000−0.91021.368E − 05Rf2−0.86973.170E − 05−0.87050.0000−0.87200.0000−0.87130.0000−0.87202.766E − 05Rf3−0.87795.507E − 05−0.87840.0001−0.88040.0001−0.87950.0001−0.88014.820E − 05150θ0.06070.02290.12440.03000.03400.00500.05040.01760.01180.0019α0.03560.0335−0.07080.0368−0.01070.00620.02130.02990.00240.0034β0.04590.0523−0.05450.0525−0.00140.01160.04600.0565−0.00580.0046Rf1−0.90851.054E − 05−0.90880.0000−0.90980.0000−0.90910.0000−0.91007.210E − 06Rf2−0.86952.049E − 05−0.86980.0000−0.87160.0000−0.87050.0000−0.87171.443E − 05Rf3−0.87783.383E − 05−0.87780.0000−0.87990.0000−0.87870.0000−0.87972.480E − 05330θ0.05710.01960.02760.01680.06440.01350.04930.01210.02120.0066α0.04590.5967−0.05330.4173−0.00950.0766−0.02480.1783−0.00220.0249β0.10030.0658−0.02670.03870.05760.05050.02140.0395−0.00910.0155Rf1−0.83260.0001−0.84590.0001−0.83240.0002−0.83740.0001−0.83860.0001Rf2−0.95940.0002−0.96270.0002−0.95930.0004−0.96060.0002−0.96090.0002Rf3−0.76600.0005−0.78520.0004−0.76580.0008−0.77290.0005−0.77480.000480θ0.03840.01550.00890.00500.01870.00520.01840.00510.00700.0030α0.03870.9714−0.03470.16970.00650.0661−0.00590.11680.00070.0194β0.05590.0538−0.03540.01340.02390.02000.00370.0150−0.00250.0077Rf1−0.83693.484E − 05−0.84300.0000−0.83660.0001−0.83830.0000−0.83863.291E − 05Rf2−0.96057.956E − 05−0.96200.0001−0.96040.0001−0.96080.0001−0.96097.884E − 05Rf3−0.77261.544E − 04−0.78140.0002−0.77220.0002−0.77460.0002−0.77521.469E − 04150θ0.01800.00490.00290.00250.02070.00310.01940.00270.00440.0015α0.00560.2288−0.02320.0877−0.01070.0529−0.01900.0501−0.00170.0080β0.02050.0145−0.01630.00790.00720.0095−0.00490.00740.00540.0040Rf1−0.83691.899E − 05−0.84092.022E − 05−0.83673.140E − 05−0.83772.241E − 05−0.83781.953E − 05Rf2−0.96054.338E − 05−0.96154.540E − 05−0.96047.036E − 05−0.96075.058E − 05−0.96074.404E − 05Rf3−0.77288.393E − 05−0.77868.635E − 05−0.77251.333E − 04−0.77409.664E − 05−0.77428.395E − 05


### 6.1. Concluding Remarks on the Simulation’s Findings

(1)
All estimation methods work admirably and have a very tiny MSE, and their mean values trend to the original values of the parameters.(2)
The discrepancies in the results of all estimators are extremely minor, both in terms of the MSE and the average of the parameters.(3)
The fuzzy reliability with high value ofγ− cut has the largest value with the largest MSE.(4)
In median sample size, we note the MPS is better than MLE.(5)
We noted that the CVM estimation method is better than MLE, MPS, and AD estimation method in Table2.(6)
We noted that AD estimation method is better than MLE, MPS, and AD estimation method in Table1.(7)
We noted that the Bayesian estimation method is better than another estimation methods.


## 6.1. Concluding Remarks on the Simulation’s Findings

(1)
All estimation methods work admirably and have a very tiny MSE, and their mean values trend to the original values of the parameters.(2)
The discrepancies in the results of all estimators are extremely minor, both in terms of the MSE and the average of the parameters.(3)
The fuzzy reliability with high value ofγ− cut has the largest value with the largest MSE.(4)
In median sample size, we note the MPS is better than MLE.(5)
We noted that the CVM estimation method is better than MLE, MPS, and AD estimation method in Table2.(6)
We noted that AD estimation method is better than MLE, MPS, and AD estimation method in Table1.(7)
We noted that the Bayesian estimation method is better than another estimation methods.


## 7. Analyses of Data Used as an Application to the Proposed Distribution

This section will demonstrate the MKKu distribution’s potential for real-world data set. The MKKu distribution is compared to many different models, including Kumaraswamy, Beta, and Gompertz Lomax (GL) (Oguntunde et al. [15]), Topp-Leone generalized exponential (TLGE) [14], Type II Power Topp-Leone invers exponential (TIIPTLIE) [13], Kumaraswamy-Kumaraswamy (KuKu) [34], alpha power inverted TL (APITL) [16], Toop-Leone Weibull Lomax (TLWL) [35], and Marshall-Olkin Kumaraswamy (MOKu) [36] distributions.Table3 contains the values of the AIC, BIC, CAIC, HQIC, Cramer-von Mises statistics (WS), Anderson–Darling statistics (AS), and Kolmogorov-Smirnov statistics (KSS), as well as their associated P-values, for all models fitted using a real data set. Additionally, these tables provide the mean and standard error (SE) of the parameters for the investigated models. Figure 3 illustrates the fit empirical, histogram, and PP-plot for the MKKu distribution for Saudi Arabia’s COVID-19 data.Table 3
MLE, SE, KS test, andP-values for COVID-19 data of Saudi Arabia.
EstimatesSEKSSP-valueAICBICCAICHQICWSAsMKKuθ2.42076.43730.07130.9782−126.6801−121.6135−126.0135−124.84820.03520.2700α1.47364.5582β5.85221.9868TIIPTLIEα1392.0427233.84710.12480.5215−124.3659−119.2993−123.6992−122.53400.09770.5808θ0.49130.0719β0.24110.2180TLGEα0.26910.32390.15070.2932−122.0071−116.9405−121.3404−120.17520.14080.8257θ18.06903.3658β118.4134181.2793Kα3.23110.27720.13510.4212−117.5307−114.1530−117.2064−116.30940.06800.4190θ125.219249.7485Betaα14.17063.13300.13400.4314−125.9165−121.5388−125.5922−124.69520.10610.6282θ54.947612.3093TLWLα1.11455.09260.07290.9730−124.8132−118.0576−123.6703−122.37060.03670.2701θ4.83084.3742β0.83090.8085λ2.723011.0853GLα3.95432.16830.07720.9563−123.5123−116.7567−122.3694−121.06970.03710.3028θ6.37663.8154β1.18290.6993λ0.00990.0075APITLα374.4046296.39690.14990.2994−118.8573−115.4795−118.5330−117.63600.11200.6662θ77.36747.3915KuKuα1.52571.27720.09160.8600−124.5031−117.7475−123.3602−122.06050.04820.3193β3.40512.2885θ22.289828.3503λ22.941233.9563MOKuθ0.00250.00060.12010.5700−118.2833−113.2166−117.6166−116.45130.14680.8665α5.86710.6903β28.237231.3740Figure 3
The cumulative function and empirical CDF, histogram, and P–P plot for MKKu distribution for COVID-19 data of Saudi Arabia.The data contains 40 days of COVID-19 data from Saudi Arabia, from 22 July to 30 August 2021 (see the link [37]). These results illustrate the drought mortality rate. The data are as follows: 0.2375 0.2962 0.2167 0.2752 0.2353 0.2347 0.1951 0.2140 0.2329 0.2711 0.2126 0.2314 0.1924 0.2113 0.2683 0.2487 0.2674 0.1716 0.2666 0.2091 0.2278 0.1706 0.2271 0.1890 0.2077 0.2452 0.1319 0.2259 0.1504 0.1879 0.1689 0.2063 0.2249 0.1686 0.1310 0.1497 0.1309 0.1495 0.1121 0.1120. The results of application of Saudi Arabia are shown in Table 3 and Figure 3.Table3 shows MLE and different measures for parameter estimates of MKKu distribution for COVID-19 data of Saudi Arabia. Table 4 shows different estimators by using different estimation methods. Figure 4 shows convergence plots of MCMC for parameter estimates of MKKu distribution. Figure 5 shows density of posterior marginal with histogram of MCMC results for parameter estimates of MKKu distribution.Table 4
Estimators by using different estimation methods.
MLECVMADBayesianEstimatesSEEstimatesSEEstimatesSEEstimatesSEθ2.42075.069624.87912.276391.97082.42072.39110.4333α1.47363.587819.61551.534663.50711.40731.46800.1240β5.852233.0466198.86066.4557532.39165.27835.84571.0195Figure 4
Trace of MCMC results and convergence of parameters for the MKKu distribution for COVID-19 data of Saudi Arabia.Figure 5
Density of posterior marginal of MCMC results of parameters for MKKu distribution for COVID-19 data of Saudi Arabia.


### 7.1. Concluding Remarks on the Real Data Application

When compared to all other models employed to fit the COVID-19 data, the MKKu distribution has the greatestP-value and the smallest distance of Kolmogorov-Smirnov (KS), W, and A values in Table 3. So we can say that the proposed distribution is superior among all its competitors as it outperforms all its rivals in fitting COVID-19 data.


## 7.1. Concluding Remarks on the Real Data Application

When compared to all other models employed to fit the COVID-19 data, the MKKu distribution has the greatestP-value and the smallest distance of Kolmogorov-Smirnov (KS), W, and A values in Table 3. So we can say that the proposed distribution is superior among all its competitors as it outperforms all its rivals in fitting COVID-19 data.


## 8. Additional Points

We propose a novel MKKu distribution generalization, in this paper. We have investigated its statistical properties and determined a linear representation for its PDF that effectively determined the linear representations of the PDF and CDF, moments, moment generating function, and fuzzy reliability function. Numerous classical and Bayesian estimation methods were considered to obtain point estimates for the unknown MKKu parametersα, β, θ. The simulation study was done by using the R package, to see and compare the performance of various estimation methods. The MCMC method was used to accomplish this, and we concluded that the Bayesian approach outperforms all the classical approaches considered. We used a COVID-19 data set from the KSA. It was determined that MKKu fitted the data more effectively than most other competing distributions.


## 9. Future Work

In future, we will work on different kinds of fuzzy reliability function under different kinds of censored data, not a complete sample to study the performance of the estimators under censoring scheme; also we will update the work to find the optimal censoring scheme. We will also go forward to apply accelerated life test on the censored data under different censoring schemes.



---

*Source: 3427521-2022-04-29.xml*