Structure of bivariate Rayleigh proportional hazard rate model with its associated copula applied on COVID‐19 data

Abstract

Visualizing the fatality of coronavirus is a very tricky point through the world. In this paper, a new construction via the proportional hazard rate model with Rayleigh marginal is introduced and applied on COVID‐19 data set. The statistical and reliability characteristics of bivariate Rayleigh proportional hazard (BRPH) distribution are derived. The copula dependence structure and its properties are studied. The point estimation of the marginal and dependence parameters is introduced via maximum likelihood, method of moments, and inference function for margins (IFM) method. A simulation study is carried out to examine the effectiveness and the performance of the parameter estimates. Finally, an application on COVID‐19 data is used in a comparison study between BRPH model and other constructed bivariate models. This application concerned with modeling the fatality on COVID‐19. Throughout the results of goodness‐of‐fit criteria, BRPH provides a better fit than different competitors constructed bivariate models which reflects its flexibility and applicability on modeling the fatality of COVID‐19.

1. INTRODUCTION

Construction of bivariate distributions has been a significant point of interest to statisticians. It mainly depends on the stochastic interpretation of the marginal distributions and the dependence structure via its copula. There are many techniques to build several bivariate distributions, see Balakrishnan and Lai. ¹ Al‐Babtain ² introduced a new extended Rayleigh distribution. In the statistical literature, the proportional hazard rate (PHR) model has a remarkable amount of devotion in constructing distributions. In 1953, Lehmann, firstly, introduced PHR model in univariate case for testing of hypothesis problem, which named Lemann alternatives. Its dual is called as the proportional reversed hazard rate (PRHR), which proposed by Gupta et al. ³

Several methods of bivariate distributions are constructed with its linked copula as Mazo et al., ⁴ Salleh et al. ⁵ and Sajid et al. ⁶ A new lifetime distribution with three‐parameter is proposed by a combination of Rayleigh distribution and extended odd Weibull family to produce the extended odd Weibull Rayleigh with applications of COVID‐19 data introduced by Almongy. ⁷

The PHR model is defined as

$$G\left(x\right)=1-{\left(1-F\left(x\right)\right)}^{\lambda },x\ge 0,\lambda >0.$$ (1)

where, $F(x)$ is the baseline cumulative distribution and $G(x)$ is the generated cumulative distribution by using PHR model.

There are more constructed models for PHR and PRHR in bivariate case; for example; Kundu and Gupta, ⁸ Mirhosseini et al., ⁹ and Sankaran and Gleeia. ¹⁰

Starting from the baseline univariate cumulative distributions $F_1(x)$ and $F_2(y),$PHR model has been proposed in bivariate case as

$$G_{\lambda }\left(x,y\right)=1-{\left(1-F_1\left(x\right)F_2\left(y\right)\right)}^{\lambda },x,y\in \mathrm{R}\mathrm{and}0<\lambda \le 1.$$

with the univariate marginal distributions

$$G_i\left(x\right)=1-{\left(1-F_i\left(x\right)\right)}^{\lambda },i=1,2.$$

where λ is the dependence parameter.

This model studied when the baseline cumulative distributions are exponential distribution functions, which is known as bivariate Kumaraswamy type exponential distribution Mirhosseini et al. ⁹ and Bakouch et al. ¹¹

Copula structure has been received a widespread attention to study the dependence structure of the bivariate distributions. In this sense, Kundu and Gupta ⁸ and Dolati et al. ¹² have studied the dependence properties of bivariate distributions in PHR and PRHR models.

Rayleigh distribution plays a vital role in several areas of research such as magnetic resonance imagining (MRI), acoustics, reliability and survival analysis, in the field of nutrition and so on. Rayleigh distribution has an important characteristic of that its hazard rate is a linear function of time. Its survival function decreases at a much higher than its correspondence with exponential distribution. The probability function of univariate Rayleigh distribution is

$$f\left(x;\alpha \right)=\frac{x}{{\alpha }^2}{e}^{-\frac{x^2}{2{\alpha }^2}},x\ge 0,\alpha >0$$

In the literature, the Morgenstern bivariate Rayleigh distribution is studied under identical parameters of the marginals, introduced in Akhter and Hirai. ¹³ The motivation of this paper is to introduce and study a new type of Bivariate Rayleigh distribution in PHR model and presenting some new constructed models of bivariate Rayleigh via different copula concepts, then compare between them and the proposed model via goodness‐of‐fit criteria.

This paper is organized as; first, the bivariate Rayleigh distribution in PHR model has been proposed. The statistical properties including the Joint PDF, the CDF, the survival function, moments, conditional moments, and correlation measures are included in Section 2. In Section 3, reliability measures are presented and discussed. Copula structure of the proposed distribution and its statistical properties are studied in Section 4. Point and interval estimations of the parameters are derived via several techniques in Section 5. In Section 6, a simulation study is proposed to study the behavior of different estimation methods. Finally, Section 7 offered an application on COVID‐19 to illustrate the potentiality of the proposed distribution compared with some other bivariate Rayleigh distribution.

2. BIVARIATE RAYLEIGH PROPORTIONAL HAZARD (BRPH) DISTRIBUTION

Let $(X_1,X_2)$ be a bivariate Rayleigh random vector with scale parameters ${\alpha }_1,{\alpha }_2$ respectively and the parameter λ, which characterize the dependence parameter. Following Dolati et al. ¹² idea of considering the Mittag–Leffler random variable, the bivariate proportional hazard rate Rayleigh joint PDF is defined as

$$g\left(x_1,x_2;{\alpha }_1,{\alpha }_2,\lambda \right)=\lambda \left(\frac{x_1}{{\alpha }_1^2}{e}^{\frac{-x_1^2}{2{\alpha }_1^2}}\right)\left(\frac{x_2}{{\alpha }_2^2}{e}^{\frac{-x_2^2}{2{\alpha }_2^2}}\right){\left(1-\left(1-e^{\frac{-x_1^2}{2{\alpha }_1^2}}\right)\left(1-e^{\frac{-x_2^2}{2{\alpha }_2^2}}\right)\right)}^{\lambda -2}\left(1-\lambda \left(1-e^{\frac{-x_1^2}{2{\alpha }_1^2}}\right)\left(1-e^{\frac{-x_2^2}{2{\alpha }_2^2}}\right)\right)$$ (2)

where, the PHR model with the baseline Rayleigh distribution

$$F_i\left(x\right)=1-e^{\frac{-x_i^2}{2{\alpha }_i^2}},i=1,2.$$

Then, its univariate cumulative distribution functions are Rayleigh distributions

$$G_i\left(x\right)=1-e^{\frac{-x_i^2\lambda }{2{\alpha }_i^2}},i=1,2.$$

The corresponding joint CDF of $(X_1,X_2)$ is

$$G\left(x_1,x_2;{\alpha }_1,{\alpha }_2,\lambda \right)=1-{\left(1-\left(1-e^{\frac{-x_1^2}{2{\alpha }_1^2}}\right)\left(1-e^{\frac{-x_2^2}{2{\alpha }_2^2}}\right)\right)}^{\lambda }$$ (3)

where x _1, $x_2\ge 0,{\alpha }_1,{\alpha }_2>0,0<\lambda \le 1$

The joint survival function is obtained as

$$S\left(x_1,x_2;{\alpha }_1,{\alpha }_2,\lambda \right)=e^{\frac{-x_1^2\lambda }{2{\alpha }_1^2}}+e^{\frac{-x_2^2\lambda }{2{\alpha }_2^2}}-{\left[1-\left(1-e^{\frac{-x_1^2}{2{\alpha }_1^2}}\right)\left(1-e^{\frac{-x_2^2}{2{\alpha }_2^2}}\right)\right]}^{\lambda }$$

Using the binomial expansion Gradshteyn and Ryzhik ¹⁴

$${(1-\beta )}^{\lambda }{\sum }_{j=0}^{\infty }\left(\begin{array}{c}\lambda \\ j\end{array}\right){(-1)}^j{\beta }^j|\beta |<1$$ (4)

After some simplifications, the joint density function for the BRPH distribution (2) can be written as

$$g\left(x_1,x_2;{\alpha }_1,{\alpha }_2,\lambda \right)=\sum _{j=1}^{\infty }\left(\begin{array}{c}\lambda \\ j\end{array}\right){\left(-1\right)}^{j+1}{j}^2\left(\frac{x_1}{{\alpha }_1^2}{e}^{\frac{-x_1^2}{2{\alpha }_1^2}}\right)\left(\frac{x_2}{{\alpha }_2^2}{e}^{\frac{-x_2^2}{2{\alpha }_2^2}}\right){\left(\left(1-e^{\frac{-x_1^2}{2{\alpha }_1^2}}\right)\left(1-e^{\frac{-x_2^2}{2{\alpha }_2^2}}\right)\right)}^{j-1}$$ (5)

Figure 1 shows two different shapes of the joint density functions of BRPH distribution with parameters $(\lambda ,{\alpha }_1,{\alpha }_2):$ $(\mathrm{A})(0.8,1,1),(\mathrm{B})(0.5,0.7,0.8),(\mathrm{C})(0.2,5,0.4),(\mathrm{D})(0.5,0.1,0.2)$. It can be seen that the densities are right‐skewed and jointly unimodal.

FIGURE 1.

Surface plots of the joint PDF of the BRPH distribution for different values of $(\lambda ,{\alpha }_1,{\alpha }_2):$ $(\mathrm{A})(0.8,1,1),(\mathrm{B})(0.5,0.7,0.8),(\mathrm{C})(0.2,5,0.4),(\mathrm{D})(0.5,0.1,0.2)$

Proposition 1. For the random vector $(X_1,X_2)\sim BRPH({\alpha }_1,{\alpha }_2,\lambda )$ then

• (1)
  For $r_1,r_2\ge 1$, the moments of $(X_1,X_2)$ is

  $$E\left(X_1^{r_1}{X}_2^{r_2}\right)=\sum _{j=1}^{\infty }\left(\begin{array}{c}\lambda \\ j\end{array}\right){\left(-1\right)}^{j+1}{j}^2\prod _{k=1}^2\frac{1}{{\alpha }_k^2}\sum _{i=0}^{j-1}\left(\begin{array}{c}j-1\\ i\end{array}\right){\left(-1\right)}^i{2}^{\frac{r_k}{2}}{\left(\frac{1+i}{{\alpha }_k^2}\right)}^{-1-\frac{r_k}{2}}\mathrm{\Gamma }\left(1+\frac{r_k}{2}\right)$$ (6)

and the Pearson's correlation coefficient of $(X_1,X_2)$ is

$$Corr\left(X_1,X_2\right)=\frac{\pi {\sum }_{j=1}^{\infty }\left(\begin{array}{c}\lambda \\ j\end{array}\right){\left(-1\right)}^{j+1}{j}^2{\prod }_{k=1}^2\left(\frac{1}{{\alpha }_k^2}{\sum }_{i=0}^{j-1}\left(\begin{array}{c}j-1\\ i\end{array}\right){\left(-1\right)}^i{\left(\frac{1+i}{{\alpha }_k^2}\right)}^{-\frac{3}{2}}\right)-{\alpha }_1{\alpha }_{2\pi }}{\left(4-\pi \right){\alpha }_1{\alpha }_2}$$ (7)

• (2)
  The moment generating function of

  $(X_1,X_2)$

  is

  $$\begin{array}{c}\hfill M_{X_1,X_2}\left(t_1,t_2\right)=\sum _{j=1}^{\infty }\left(\begin{array}{c}\lambda \\ j\end{array}\right){\left(-1\right)}^{j+1}{j}^2\prod _{k=1}^2\left(\frac{1}{{\alpha }_k^2}\sum _{i=0}^{j-1}\left(\begin{array}{c}j-1\\ i\end{array}\right){\left(-1\right)}^i\frac{{\mathrm{e}}^{\frac{t_k^2{\alpha }_k^2}{2\left(1+i\right)}}\sqrt{\pi }(1+\mathrm{Erf}\left({\left(\frac{2\left[i+1\right]}{{\alpha }_k^2}\right)}^{\frac{-1}{2}}{t}_k\right)+{\left(\frac{2\left(1+i\right)}{{\alpha }_k^2}\right)}^{\frac{1}{2}}}{\sqrt{2{\left(\frac{1+i}{{\alpha }_k^2}\right)}^3}{\alpha }_k^2}\right)\end{array}$$ (8)

where $Erf(x)=\frac{2}{\sqrt{\pi }}\underset{0}{\overset{x}{\int }}{e}^{-t^2}dt$ is the error function.

Proof.

For part (1), using Equation (5)

$$E\left(X_1^{r_1}{X}_2^{r_2}\right)=\sum _{j=1}^{\infty }\left(\begin{array}{c}\lambda \\ j\end{array}\right){\left(-1\right)}^{j+1}{j}^2\prod _{k=1}^2{I}_k\left(X_k,r_k\right)$$

where

$$I_k\left(X_k,r_k\right)=\underset{0}{\overset{\infty }{\int }}\frac{x_k^{r_k+1}}{{\alpha }_k^2}{e}^{\frac{-x_k^2}{2{\alpha }_k^2}}{\left(1-e^{\frac{-x_k^2}{2{\alpha }_k^2}}\right)}^{j-1}d{x}_k\text{for}k=1,2$$

By using the expansion (4), then

$$I_k\left(X_k,r_k\right)=\frac{1}{{\alpha }_k^2}\sum _{i=0}^{j-1}\left(\begin{array}{c}j-1\\ i\end{array}\right){\left(-1\right)}^i\underset{0}{\overset{\infty }{\int }}{x}_k^{r_k+1}{e}^{\frac{-x_k^2\left(i+1\right)}{2{\alpha }_k^2}}d{x}_k$$

and $\underset{0}{\overset{\infty }{\int }}{x}_k^{r_k+1}{e}^{\frac{-x_k^2(i+1)}{2{\alpha }_k^2}}d{x}_k=2^{\frac{r_k}{2}}{(\frac{1+i}{{\alpha }_k^2})}^{-1-\frac{r_k}{2}}\mathrm{\Gamma }(1+\frac{r_k}{2})$

Hence, the expression for the product moment is proved.

For part (2),

$$M_{X_1,X_2}\left(t_1,t_2\right)=E\left[e^{t_1{x}_1+t_2{x}_2}\right]$$

$$\begin{array}{c}\hfill =\sum _{j=1}^{\infty }\left(\begin{array}{c}\lambda \\ j\end{array}\right){\left(-1\right)}^{j+1}{j}^2\prod _{k=1}^2{M}_k\left(X_k,t_k\right)\end{array}$$

$$M_k\left(X_k,t_k\right)=\underset{0}{\overset{\infty }{\int }}{e}^{t_k{x}_k}\frac{x_k^{r_k}}{{\alpha }_k^2}{e}^{\frac{-x_k^2}{2{\alpha }_k^2}}{\left(1-e^{\frac{-x_k^2}{2{\alpha }_k^2}}\right)}^{j-1}d{x}_k$$

which consequently implies

$$M_k\left(X_k,t_k\right)=\frac{1}{{\alpha }_k^2}\sum _{i=0}^{j-1}\left(\begin{array}{c}j-1\\ i\end{array}\right){\left(-1\right)}^i\underset{0}{\overset{\infty }{\int }}{x}_k{e}^{t_k{x}_k}{e}^{\frac{-x_k^2\left(i+1\right)}{2{\alpha }_k^2}}d{x}_k$$

and,

$$\underset{0}{\overset{\infty }{\int }}{x}_k{e}^{t_k{x}_k}{e}^{\frac{-x_k^2\left(i+1\right)}{2{\alpha }_k^2}}d{x}_k=\frac{{\mathrm{e}}^{\frac{t_k^2{\alpha }_k^2}{2\left(1+i\right)}}\sqrt{\pi }(1+\mathrm{Erf}\left({\left(\frac{2\left[i+1\right]}{{\alpha }_k^2}\right)}^{\frac{-1}{2}}{t}_k\right)+{\left(\frac{2\left(1+i\right)}{{\alpha }_k^2}\right)}^{\frac{1}{2}}}{\sqrt{2{\left(\frac{1+i}{{\alpha }_k^2}\right)}^3}{\alpha }_k^2}$$

Some numerical values of Pearson correlation coefficient values are obtained in Table 1. It is noticed that at small values of the dependence parameter λ, the correlation coefficient is negative weak, then it changes to be positive tends to be strong positive correlation as long as λ approaches 0.5, then it declines till zero at $\lambda =1$ where the independency fulfills.

TABLE 1.

Correlation coefficient for some values of α₁,${\alpha }_2\mathrm{and}$ λ

α1	α2	λ
		Correlation coefficient
0.01	0.01	0.2	0.3	0.5	0.7	0.9	1
		−0.62144	0.18439	0.86537	0.774582	0.30074	0
0.05	0.07	0.21	0.31	0.51	0.71	0.91	1
		−0.52302	0.244949	0.875209	0.757469	0.271821	0
0.1	0.2	0.22	0.32	0.52	0.72	0.92	1
		−0.52302	0.302241	0.883214	0.739467	0.24259	0
0.3	0.4	0.23	0.33	0.53	0.73	0.93	1
		−0.33876	0.35635	0.889448	0.72061	0.21307	0
0.5	0.5	0.24	0.34	0.54	0.74	0.94	1
		−0.25274	0.40736	0.893967	0.700936	0.18328	0
0.65	0.75	0.25	0.35	0.55	0.75	0.95	1
		−0.17064	0.455356	0.896828	0.680479	0.153242	0
0.8	1	0.26	0.36	0.56	0.76	0.96	1
		−0.09236	0.500417	0.898082	0.659272	0.122976	0
1.5	3.8	0.27	0.37	0.57	0.77	0.97	1
		−0.01782	0.542626	0.897784	0.637348	0.0925	0
15.2	16.8	0.28	0.38	0.58	0.78	0.98	1
		0.05309	0.58206	0.895985	0.61474	0.061833	0
50	100	0.29	0.39	0.59	0.79	0.99	1
		0.120465	0.618796	0.892735	0.59148	0.0309942	0

The conditional density function of $X_2$given $X_1=t$ is obtained in the form

$$g_{X_2/X_1=t}\left(x_2\right)=\frac{x_2}{\lambda {\alpha }_2^2}{e}^{\frac{-1}{2{\alpha }_1^2}\left(1-\lambda \right)t^2-\frac{{x_2}^2}{2{\alpha }_2^2}}\sum _{j=1}^{\infty }\left(\begin{array}{c}\lambda \\ j\end{array}\right){\left(-1\right)}^{j+1}{j}^2{\left(\left(1-e^{\frac{-t^2}{2{\alpha }_1^2}}\right)\left(1-e^{\frac{-x_2^2}{2{\alpha }_2^2}}\right)\right)}^{j-1}$$ (9)

Proposition 2. For the random vector $(X_1,X_2)\sim BRPH({\alpha }_1,{\alpha }_2,\lambda ),$ then the conditional moment of order r is

$$E\left(X_2^r/X_1=t\right)=\frac{1}{\lambda {\alpha }_2^2}{e}^{\frac{-1}{2{\alpha }_1^2}\left(1-\lambda \right)t^2}\sum _{j=1}^{\infty }\left(\begin{array}{c}\lambda \\ j\end{array}\right){\left(-1\right)}^{j+1}{j}^2{\left(1-e^{\frac{-t^2}{2{\alpha }_1^2}}\right)}^{j-1}\sum _{i=0}^{j-1}\left(\begin{array}{c}j-1\\ i\end{array}\right){\left(-1\right)}^i{\left(2\right)}^{\frac{r}{2}}{\left(\frac{1+i}{{\alpha }_2^2}\right)}^{-1-\frac{r}{2}}\mathrm{\Gamma }\left(1+\frac{r}{2}\right)$$ (10)

3. RELIABILITY MEASURES

3.1. Stress–strength parameter

It is essential to take into account the stress–strength parameter in measuring the reliability of the operating systems. The system is reliable as long as the stress X ₁ is less than the strength $X_2$.

Proposition 3. Assuming that X ₁ and $X_2$ are jointly distributed as $BRPH({\alpha }_1,{\alpha }_2,\lambda )$, the stress–strength parameter $R=P(X_1<X_2)$ is obtained as

$$P\left(X_1<X_2\right)=\sum _{j=1}^{\infty }\left(\begin{array}{c}\lambda \\ j\end{array}\right){\left(-1\right)}^{j+1}j\sum _{i=0}^{j-1}\left(\begin{array}{c}j-1\\ i\end{array}\right){\left(-1\right)}^i\sum _{k=0}^j\left(\begin{array}{c}j\\ k\end{array}\right){\left(-1\right)}^k\frac{{\alpha }_1^2}{\left(1+i\right){\alpha }_1^2+k{\alpha }_2^2}$$ (11)

3.2. The hazard rate function

The bivariate hazard rate is defined in several ways in the literature. One is due to Basu ¹⁵ as

$$h\left(x_1,x_2\right)=\frac{g\left(x_1,x_2\right)}{S\left(x_1,x_2\right)}$$

According to this definition, the hazard rate function for $BRPH$ distribution is

$$h\left(x_1,x_2\right)=\frac{{\mathrm{e}}^{\frac{-x_1^2}{2{\alpha }_1^2}-\frac{x_2^2}{2{\alpha }_2^2}}{\left(1-\left(1-{\mathrm{e}}^{\frac{-x_1^2}{2{\alpha }_1^2}}\right)\left(1-{\mathrm{e}}^{\frac{-x_2^2}{2{\alpha }_2^2}}\right)\right)}^{-2+\lambda }{x}_1{x}_2\lambda \left(1-\left(1-{\mathrm{e}}^{-\frac{x_1^2}{2{\alpha }_1^2}}\right)\left(1-{\mathrm{e}}^{\frac{-x_2^2}{2{\alpha }_2^2}}\right)\lambda \right)}{\left({\mathrm{e}}^{\frac{-x_1^2\lambda }{2{\alpha }_1^2}}+{\mathrm{e}}^{\frac{-x_2^2\lambda }{2{\alpha }_2^2}}-{\left(1-\left(1-{\mathrm{e}}^{\frac{-x_1^2}{2{\alpha }_1^2}}\right)\left(1-{\mathrm{e}}^{\frac{-x_2^2}{2{\alpha }_2^2}}\right)\right)}^{\lambda }\right){\alpha }_1^2{\alpha }_2^2}$$ (12)

Another point of view, Johnson and Kotz ¹⁶ define it in a vector form by the following rule

$$h\left(x_1,x_2\right)=\left(\frac{-\partial \ln S\left(x_1,x_2\right)}{\partial x_1},\frac{-\partial \ln S\left(x_1,x_2\right)}{\partial x_2}\right)$$

Therefore, the hazard for BRPH distribution can be defined by

$$\frac{-\partial \ln S\left(x_1,x_2\right)}{\partial x_1}=-\frac{\left(-{\mathrm{e}}^{-\frac{\lambda x_1^2}{2{\alpha }_1^2}}+{\mathrm{e}}^{-\frac{x_1^2}{2{\alpha }_1^2}}\left(1-{\mathrm{e}}^{-\frac{x_2^2}{2{\alpha }_2^2}}\right){\left(1-\left(1-{\mathrm{e}}^{-\frac{x_1^2}{2{\alpha }_1^2}}\right)\left(1-{\mathrm{e}}^{-\frac{x_2^2}{2{\alpha }_2^2}}\right)\right)}^{-1+\lambda }\right)\lambda x_1}{\left({\mathrm{e}}^{-\frac{\lambda x_1^2}{2{\alpha }_1^2}}+{\mathrm{e}}^{-\frac{\lambda x_2^2}{2{\alpha }_2^2}}-{\left(1-\left(1-{\mathrm{e}}^{-\frac{x_1^2}{2{\alpha }_1^2}}\right)\left(1-{\mathrm{e}}^{-\frac{x_2^2}{2{\alpha }_2^2}}\right)\right)}^{\lambda }\right){\alpha }_1^2}$$

and, $\frac{-\partial \ln S(x_1,x_2)}{\partial x_2}=-\frac{(-{\mathrm{e}}^{-\frac{\lambda x_2^2}{2{\alpha }_2^2}}+{\mathrm{e}}^{-\frac{x_2^2}{2{\alpha }_2^2}}(1-{\mathrm{e}}^{-\frac{x_1^2}{2{\alpha }_1^2}}){(1-(1-{\mathrm{e}}^{-\frac{x_1^2}{2{\alpha }_1^2}})(1-{\mathrm{e}}^{-\frac{x_2^2}{2{\alpha }_2^2}}))}^{-1+\lambda })\lambda x_2}{({\mathrm{e}}^{-\frac{\lambda x_1^2}{2{\alpha }_1^2}}+{\mathrm{e}}^{-\frac{\lambda x_2^2}{2{\alpha }_2^2}}-{(1-(1-{\mathrm{e}}^{-\frac{x_1^2}{2{\alpha }_1^2}})(1-{\mathrm{e}}^{-\frac{x_2^2}{2{\alpha }_2^2}}))}^{\lambda }){\alpha }_2^2}$

3.3. The mean residual life (MRL) and the vitality function

The MRL is defined as the average remaining time of the system after it has survived for a specified time t. Shanbag and Kotz ¹⁷ and Vaidyanathan and Varghese ¹⁸ defined it in a vector form as

$$MRL\left(x_1,x_2\right)=\left(M_1\left(x_1,x_2\right),M_2\left(x_1,x_2\right)\right)$$

where

$$M_1\left(x_1,x_2\right)=E\left(X_1-x_1/X_1\ge x_1,X_2\ge x_2\right)$$

$$M_2\left(x_1,x_2\right)=E\left(X_2-x_2/X_2\ge x_2,X_1\ge x_1\right)$$

Hence, the MRL for $BRPH$ distribution is

$$M_1\left(x_1,x_2\right)=\frac{1}{S\left(x_1,x_2\right)}\sum _{j=1}^{\infty }\left(\begin{array}{c}\lambda \\ j\end{array}\right){\left(-1\right)}^{j+1}{j}^2{I}_1{I}_2-x_1$$ (13)

where

$$I_1=\sum _{i=0}^{j-1}\left(\begin{array}{c}j-1\\ i\end{array}\right){\left(-1\right)}^i\frac{{\mathrm{e}}^{-\frac{\left(1+i\right)x_2^2}{2{\alpha }_2^2}}}{1+i}$$

$$I_2=\sum _{i=0}^{j-1}\left(\begin{array}{c}j-1\\ i\end{array}\right){\left(-1\right)}^i\frac{\left(2{\mathrm{e}}^{-\frac{\left(1+i\right)x_1^2}{2{\alpha }_1^2}}{x}_1+{\frac{2\pi \left({\alpha }_1^2\right)}{1+i}}^{\frac{1}{2}}-\frac{{\alpha }_1\sqrt{2\pi }\mathrm{Erf}\left[\frac{x_1}{{\alpha }_1}{\left(\frac{1+i}{2}\right)}^{\frac{1}{2}}\right]}{\sqrt{1+i}}\right)}{2\left(1+i\right)}$$

$$M_2\left(x_1,x_2\right)=\frac{1}{S\left(x_1,x_2\right)}\sum _{j=1}^{\infty }\left(\begin{array}{c}\lambda \\ j\end{array}\right){\left(-1\right)}^{j+1}{j}^2{I}_3{I}_4-x_2$$

where

$$I_3=\sum _{i=0}^{j-1}\left(\begin{array}{c}j-1\\ i\end{array}\right){\left(-1\right)}^i\frac{{\mathrm{e}}^{-\frac{\left(1+i\right)x_1^2}{2{\alpha }_1^2}}}{1+i}$$

$$I_4=\sum _{i=0}^{j-1}\left(\begin{array}{c}j-1\\ i\end{array}\right){\left(-1\right)}^i\frac{\left(2{\mathrm{e}}^{-\frac{\left(1+i\right)x_2^2}{2{\alpha }_2^2}}{x}_2+{\frac{2\pi \left({\alpha }_2^2\right)}{1+i}}^{\frac{1}{2}}-\frac{{\alpha }_2\sqrt{2\pi }\mathrm{Erf}\left[\frac{x_2}{{\alpha }_2}{\left(\frac{1+i}{2}\right)}^{\frac{1}{2}}\right]}{\sqrt{1+i}}\right)}{2\left(1+i\right)}$$

In the case of bivariate components of the system, the bivariate vitality function is related to MRL as

$$v\left(x_1,x_2\right)=\left(v_1\left(x_1,x_2\right),v_2\left(x_1,x_2\right)\right)$$

$$v_i\left(x_1,x_2\right)=x_i+M_i\left(x_1,x_2\right),i=1,2$$

and it can be calculated immediately.

4. COPULA STRUCTURE AND DEPENDENCE PROPERTIES

Copula is a way to construct bivariate models with a variety of dependence structures via Sklar's theorem Sklar ¹⁹ by solving the equation

$$C_{\lambda }\left(F_1\left(x_1\right),F_2\left(x_2\right)\right)=G\left(x_1,x_2\right)$$ (14)

For $C_{\lambda }:[0,1]\times [0,1]\to [0,1]$. Sklar's theorem supported us to present dependence properties of the $BRPH$ distribution through its associated copula. The function $C_{\lambda }(F_1(x_1),F_2(x_2))$ can be written as $C_{\lambda }(u,v)$ which $u,v$ is a distribution function of Rayleigh distribution, which produces the following three equivalent equations associated with $G(x_1,x_2)$:

$$C_{\lambda }\left(u,v\right)=1-{\left\{1-\left(1-{\left(1-u\right)}^{\frac{1}{\lambda }}\right)\left(1-{\left(1-v\right)}^{\frac{1}{\lambda }}\right)\right\}}^{\lambda },$$ (15)

or

$$C_{\lambda }\left(u,v\right)=1-\left(1-u\right)\left(1-v\right){\left\{{\left(1-u\right)}^{-\frac{1}{\lambda }}+{\left(1-v\right)}^{-\frac{1}{\lambda }}-1\right\}}^{\lambda },$$

or

$$C_{\lambda }\left(u,v\right)=1-{\left[{\left(1-u\right)}^{\frac{1}{\lambda }}+{\left(1-v\right)}^{\frac{1}{\lambda }}-{\left(1-u\right)}^{\frac{1}{\lambda }}{\left(1-v\right)}^{\frac{1}{\lambda }}\right]}^{\lambda },$$

for all $u,v\in (0,1)$ and $0<\lambda \le 1$, which is one of the Archimedean family of copulas with strict generator $\varphi (t)=-\ln [1-{(1-t)}^{\frac{1}{\lambda }}]$ for more detail see, Nelsen. ²⁰

The density function of the copula is $c_{\lambda }(u,v)=\frac{{\partial }^2}{\partial u\partial v}{C}_{\lambda }(u,v)$, so the density function for the copula in Equation (15) is denoted by

$$c_{\lambda }\left(u,v\right)=\lambda \left[1-\lambda \left(1-{\left(1-u\right)}^{\frac{1}{\lambda }}\right)\left(1-{\left(1-v\right)}^{\frac{1}{\lambda }}\right)\right]{\left[1-\left(1-{\left(1-u\right)}^{\frac{1}{\lambda }}\right)\left(1-{\left(1-v\right)}^{\frac{1}{\lambda }}\right)\right]}^{\lambda -2}$$ (16)

Now several properties for $G(x_1,x_2)$ in terms of copula $C_{\lambda }(u,v)$ will be studied as concordance ordering, tail monotonicity, measures of association and symmetry.

• Concordance ordering

For two copulas C ₁ and C ₂, we can say that C ₂ is more concordant than C ₁ (written $C_1{\prec }_c{C}_2)$ if $C_1\le C_2$, for all $u,v\in (0,1)$. The copula $C_{\lambda }(u,v)$ given by Equation (15) is negatively ordered with respect to λ. A copula C is positively quadrant‐dependent (PQD) if $\mathrm{\Pi }{\prec }_{c}C$, where $\mathrm{\Pi }=uv$ is the product copula, for details see Nelsen. ²⁰ As a result, a pair $(X_1,X_2)$ distributed as $BRPH$ is PQD, where $\mathrm{\Pi }=C_1(u,v)\le C_{\lambda }(u,v)$.

• Tail monotonicity

Let $(X_1,X_2)$ be a BRPH continuous random variables whose associated copula is $C_{\lambda }(u,v)$ given in Equation (15), then X ₂ is left tail decreasing in X ₁ (LTD($X_2|X_1)$) where $\frac{\partial C_{\lambda }(u,v)}{\partial u}\le \frac{C_{\lambda }(u,v)}{u}$ for almost all u, and also X ₁ is left tail decreasing in X ₂ (LTD($X_1|X_2)$) where $\frac{\partial C_{\lambda }(u,v)}{\partial v}\le \frac{C_{\lambda }(u,v)}{v}$ for almost all v, holds in $\mathrm{I}=(0,1)$. By routine calculations, we can obtain X ₁ is right tail increasing in X ₂ (RTI($X_1|X_2)$) in $\mathrm{I}$ holds where $\frac{\partial C_{\lambda }(u,v)}{\partial v}\ge \frac{[u-C_{\lambda }(u,v)]}{(1-v)}$ for almost all v. If X ₁ and X ₂ are left corner set decreasing (LCSD($X_1,X_2$)), then (LTD($X_2|X_1)$) and (LTD($X_1|X_2)$) holds see Corollary 5.2.17 in Nelsen. ²⁰ The following flowchart introduces the relationship among several dependence properties for BRPH distribution

$$\begin{array}{c}(\mathbf{LTD}({\mathit{X}}_{\mathbf{2}}\left|{\mathit{X}}_{\mathbf{1}})\right.)\Rightarrow \mathbf{PQD}(({\mathit{X}}_{\mathbf{1}}\left|{\mathit{X}}_{\mathbf{2}})\right.)\Leftarrow (\mathbf{RTI}(({\mathit{X}}_{\mathbf{1}}\left|{\mathit{X}}_{\mathbf{2}})\right.)\hfill \\ \Uparrow \Uparrow \hfill \\ (\mathbf{LCSD}({\mathit{X}}_{\mathbf{1}},{\mathit{X}}_{\mathbf{2}}))\Rightarrow (\mathbf{LTD}({\mathit{X}}_{\mathbf{1}},{\mathit{X}}_{\mathbf{2}}))\hfill \end{array}$$

• Measures of association

Dependence properties and measures of association are interrelated so four nonparametric measures of association between a continuous random pair ($X_1,X_2$) will be introduced such as Kendall's tau (τ), Spearman's rho (ρ), Spearman's footrule coefficient (${\phi }_C$), and Blomqvist medial correlation coefficient (β), which depends only on the copula C and are given by

$$\tau =4\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}C\left(u,v\right)dC\left(u,v\right)-1,$$ (17)

$$\rho =12\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}C\left(u,v\right)dudv-3,$$ (18)

Due to Dolati et al., ¹² the following proposition is summarized

Proposition 4: Let $C_{\lambda }(u,v)$ be a copula defined in Equation (15) then for every $0<\lambda \le 1$

$$\tau =1+4\lambda B\left(2,2\lambda -1\right)\left(\mathrm{\Psi }\left(2\right)-\mathrm{\Psi }\left(2\lambda +1\right)\right),$$

$$\rho =9-12{\lambda }^2\sum _{j=0}^{\infty }{\left(-1\right)}^j\left(\begin{array}{c}\lambda \\ j\end{array}\right){\left[B\left(j+1,\lambda \right)\right]}^2,$$

where Ψ is the digamma function,

Moreover, Spearman's footrule coefficient ${\phi }_C$can be calculated after some integration as follows:

$${\phi }_C=6\underset{0}{\overset{1}{\int }}C\left(u,u\right)du-2,$$

Hence, ${\phi }_C=6[1-8^{\lambda }\lambda B[\frac{1}{2},2\lambda ,1+\lambda ]]-2$

The medial correlation coefficient for a pair BRPH ($X_1,X_2$) is defined as

$$M_{XY}=P\left[\left(X_1-M_{X_1}\right)\left(X_2-M_{X_2}\right)>0\right]-P\left[\left(X_1-M_{X_1}\right)\left(X_2-M_{X_2}\right)<0\right]$$

where $M_{X_1}$ and $M_{X_2}$ are medians of X ₁ and X ₂. Some numerical values for medial correlation coefficient are given in Table 2. Using the copula function, the medial correlation becomes Blomqvist medial correlation coefficient (β)

$$\beta =4C\left(\frac{1}{2},\frac{1}{2}\right)-1,$$

TABLE 2.

Medial correlation associated with BRPH distribution for different values of λ

λ	0.02	0.03	0.05	0.08	0.1	0.15	0.2
Blomqvist medial correlation (β)	0.972	0.958	0.9295	0.886	0.857	0.783	0.709
λ	0.25	0.4	0.45	0.5	0.65	0.7	0.75	0.95
Blomqvist medial correlation (β)	0.6404	0.4569	0.4038	0.354	0.2243	0.1863	0.1506	0.0269

Therefore, $\beta =3-{(2\frac{1+\lambda }{\lambda }-1)}^{\lambda }$.

5. PARAMETER ESTIMATION

In this section, the estimation of parameters by the maximum likelihood estimation, method of moments, and the inference function for margins (IFM) method will be introduced based on random samples $(x_{1i},x_{2i}),i=1,2,\text{…},n$ from $BRPH$ distribution with parameters ${\alpha }_1,{\alpha }_2,$ and λ.

5.1. Maximum likelihood estimation

The log likelihood function $\log L$for the BRPH parameter vector $\mathrm{\Theta }={({\alpha }_1,{\alpha }_2,\lambda )}^T$has obtained as

$$\begin{array}{ccc}\hfill \log L& =& n\log \lambda +{\sum }_{i=1}^n\log \left(\frac{x_{1i}}{{\alpha }_1^2}{e}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\right)+{\sum }_{i=1}^n\log \left(\frac{x_{2i}}{{\alpha }_2^2}{e}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)+(\lambda -2){\sum }_{i=1}^n\log \left(1-\left(1-e^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\right)\left(1-e^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)\right)\hfill \\ & & +{\sum }_{i=1}^n\log \left(1-\lambda \left(1-e^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\right)\left(1-e^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)\right)\hfill \end{array}$$ (19)

The maximum likelihood estimates can be obtained by maximizing Equation (19) w.r.to the unknown parameters. The normal equations are given by

$$\begin{array}{ccc}\hfill \frac{\partial \log L}{\partial {\alpha }_1}& =& {\sum }_{i=1}^n\left(\frac{x_{1i}^2}{{\alpha }_1^3}-\frac{2}{{\alpha }_1}\right)+\left(\lambda -2\right){\sum }_{i=1}^n\frac{{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)x_{1i}^2}{\left(1-\left(1-{\mathrm{e}}^{-\frac{x}{2{\alpha }_1^2}}\right)\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)\right){\alpha }_1^3}\hfill \\ & & +{\sum }_{i=1}^n\frac{{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)\lambda x_{1i}^2}{\left(1-\left(1-{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\right)\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)\lambda \right){\alpha }_1^3}=0\hfill \end{array}$$ (20)

$$\begin{array}{ccc}\hfill \frac{\partial \log L}{\partial {\alpha }_2}& =& \sum _{i=1}^n\left(\frac{x_{2i}^2}{{\alpha }_2^3}-\frac{2}{{\alpha }_2}\right)+\left(\lambda -2\right)\sum _{i=1}^n\frac{{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\left(1-{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\right)x_{2i}^2}{\left(1-\left(1-{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\right)\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)\right){\alpha }_2^3}\hfill \\ & & +\sum _{i=1}^n\frac{{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\left(1-{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\right)\lambda x_{2i}^2}{\left(1-\left(1-{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\right)\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)\lambda \right){\alpha }_2^3}=0\hfill \end{array}$$ (21)

$$\frac{\partial \log L}{\partial \lambda }=\frac{m}{\lambda }+\sum _{i=1}^n-\frac{\left(1-{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\right)\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)}{\left(1-\left(1-{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\right)\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)\lambda \right)}+\sum _{i=1}^n\log \left(1-\left(1-{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\right)\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)\right)=0$$ (22)

The previous equations are nonlinear with respect to the parameters and cannot be obtained in explicit forms, so it can be solved numerically using the initial guesses of the parameters. The initial value for the association parameter λ, say, $\widehat{\lambda }$ can be computed using one of the sample estimates of Kendall's tau ${\tau }_n$ or Spearman rho ${\rho }_n$. Since ${\tau }_n$ and ${\rho }_n$ cannot be calculated theoretically from Equations (17) and (18), the numerical solution could be used. Also, the initial estimates for the parameters α₁ and α₂ will be obtained by making the reparameterization: $\frac{1}{{\beta }_1^2}=\frac{\lambda }{{\alpha }_1^2}$, $\frac{1}{{\beta }_2^2}=\frac{\lambda }{{\alpha }_2^2}$. Since $X_1\sim Ray({\beta }_1)$ and $X_2\sim Ray({\beta }_2)$, the estimates ${\widehat{\beta }}_1$ and ${\widehat{\beta }}_2$ can be calculated based on the marginals and then put ${\widehat{\alpha }}_1=\sqrt{\widehat{\lambda }{\widehat{\beta }}_1^2}$ and ${\widehat{\alpha }}_2=\sqrt{\widehat{\lambda }{\widehat{\beta }}_2^2}$ as the initial estimates of α₁ and α₂.

Since BRPH is a regular family, the asymptotic normality results holds, that is, $\sqrt{n}(\widehat{\theta }-\theta ){\to }^d{N}_3(0,I^{-1})$, where I is the Fisher information matrix. The observed Fisher information matrix is obtained (in the Appendix), which can be used to obtain the asymptotic confidence intervals of the unknown parameters.

5.2. Method of moments

Let the population moment for j th variate is ${\mu }_j=E(X_j),j=1,2$, the sample moment and the sample joint moment are respectively $m_j=\frac{{\sum }_{i=1}^n{X}_{ji}}{n}$, $\overline{X_1{X}_2}=\sum _{i=1}^n\frac{X_{1i}{X}_{2i}}{n}$.

The method of moments of marginal parameters ${\alpha }_1,{\alpha }_{2,}$ and λ, say, ${\widehat{\alpha }}_1,{\widehat{\alpha }}_2,$ and $\widehat{\lambda }$ for $BRPH$ distribution can be calculated by solving the following equations for the sample moment and the sample joint moment, respectively:

$$m_j=E\left(X_j\right)={\alpha }_j\sqrt{\frac{\pi }{2\lambda }},j=1,2$$

$$\overline{X_1{X}_2}=E\left(X_1{X}_2\right)=\sum _{j=1}^{\infty }\left(\begin{array}{c}\lambda \\ j\end{array}\right){\left(-1\right)}^{j+1}{j}^2\prod _{k=1}^2\frac{1}{{\alpha }_k^2}\sum _{i=0}^{j-1}\left(\begin{array}{c}j-1\\ i\end{array}\right){\left(-1\right)}^i\sqrt{\frac{\pi }{2}}{\left(\frac{1+i}{{\alpha }_k^2}\right)}^{\frac{-3}{2}}$$ (23)

5.3. Inference function for margins (IFM)

IFM method introduced by Joe and Xu ²¹ are used for estimating parameters for multivariate models when each of the parameters either marginal of the dependent parameter of the model can be related with a marginal distribution. This method is computed by estimating the univariate parameters by maximizing their corresponding univariate likelihoods and estimating the dependence parameter by maximizing the bivariate likelihood, then the estimates, ${\widehat{\alpha }}_1,{\widehat{\alpha }}_2,$ and $\widehat{\lambda }$ are obtained by solving the following log‐likelihood equations. Let $L_j$, $j=1,2$ be the log‐likelihood function of X ₁ and X ₂.

$$\frac{\partial \log L_j}{\partial {\alpha }_j}=\frac{-2n}{{\alpha }_j}+\lambda \frac{{\sum }_{i=1}^n{x}_{ji}^2}{{\alpha }_j^3}=0$$

$${\widehat{\alpha }}_j={\left(\lambda \frac{{\sum }_{i=1}^n{x}_{ji}^2}{2n}\right)}^{1/\phantom{122}2},j=1,2$$ (24)

Then, using the Equation (22) for joint likelihood Equation and solving with Equation (24), the estimates can be found much easier. IFM method is simpler than other methods of estimation which estimating all parameters from the joint likelihood function.

6. SIMULATION STUDY

The simulation study is carried out to examine the performance of the parameter estimates, which obtained via the mentioned parameter estimation methods. The simulation study was repeated 1000 times each with sample sizes; $n=30,50,80,100$ and the nodes are $({\alpha }_1,{\alpha }_2,\lambda )=(0.3,0.4,0.2),(0.4,0.1,0.5)$. Average bias and mean square error (MSE) of the parameter estimates are computed for each step. The results are given in Table 3.

TABLE 3.

Results of the simulation comparison study

n	MLE method
	$MSE({\alpha }_1)$	$Bias({\alpha }_1)$	$MSE({\alpha }_2)$	$Bias({\alpha }_2)$	$MSE(\lambda )$	$Bias(\lambda )$	Initials ${\alpha }_1=0.3,$ ${\alpha }_2=0.4,$ $\lambda =0.2$
30	0.03596	0.17219	0.00446	0.00568	0.5807	0.6756
50	0.0329	0.17166	0.002373	0.010145	0.4634	0.6430
80	0.0298	0.1676	0.00129	0.013076	0.4068	0.621034
100	0.03021	0.1696	0.0008137	0.01398	0.39305	0.616357
	MME method
30	0.006637	0.0783	0.04909	0.220514	0.0557	0.222	Initials ${\alpha }_1=0.3,$ ${\alpha }_2=0.4,$ $\lambda =0.2$
50	0.00629	0.0772	0.04842	0.2194	0.0611	0.2409
80	0.00623	0.0778	0.0487	0.2203	0.0638	0.2491
100	0.0061	0.0770	0.0482	0.2192	0.0653	0.2534

n	IFM method
30	0.005692	0.07382	0.04405	0.209302	11.0603	2.6705	Initials ${\alpha }_1=0.3,$ ${\alpha }_2=0.4,$ $\lambda =0.2$
50	0.005588	0.07380	0.004415	0.209784	8.76662	2.35189
80	0.005426	0.07301	0.043767	0.208988	5.35891	1.83346
100	0.005446	0.07327	0.04398	0.20956	3.29645	1.509

n	MLE method
	$MSE({\alpha }_1)$	$Bias({\alpha }_1)$	$MSE({\alpha }_2)$	$Bias({\alpha }_2)$	$MSE(\lambda )$	$Bias(\lambda )$	Initials ${\alpha }_1=0.4,$ ${\alpha }_2=0.1,$ $\lambda =0.5$
30	0.009708	0.0594	0.2178	0.3955	0.26405	0.29741
50	0.00738	0.0437	0.232	0.4287	0.2346	0.32976
80	0.00708	0.037	0.2306	0.4326	0.21108	0.3252
100	0.0062	0.0339	0.2276	0.4348	0.19213	0.31995
	MME method
30	0.01474	0.116705	0.154039	0.391434	0.0014404	0.0379519	Initials ${\alpha }_1=0.4,$ ${\alpha }_2=0.1,$ $\lambda =0.5$
50	0.013959	0.11501	0.155406	0.393614	0.014403	0.0379516
80	0.01388	0.116079	0.154887	0.393178	0.0144032	0.0379515
100	0.01357	0.114891	0.155785	0.3944	0.0140034	0.0379518

n	IFM method
30	0.0101822	0.097669	0.128217	0.357701	8.04508	2.0504	Initials ${\alpha }_1=0.4,$ ${\alpha }_2=0.1,$ $\lambda =0.5$
50	0.0097606	0.09703	0.127723	0.357182	3.97463	1.33954
80	0.009836	0.098043	0.128428	0.358233	0.4584	0.657558
100	0.009966	0.098899	0.127668	0.351799	0.370286	0.602852

The note points on the simulation results can be summarized as follows:

• The biases and MSE of MLE, MME, and IFM approaches mostly decrease as sample size increases.

• The convergence of the bias and MSE to zero appears very slowly especially for ${\alpha }_1,{\alpha }_2$ w.r.to the three approaches.

• The MLE appears the most stable in decaying to zero for the bias and MSE, follows by MME approach, which sometimes tends to change slightly.

• MLE and MME provide better results to estimate the parameter$\lambda$ closer to the initials rather than the IFM approach.

• The MSEs and biases seem smallest for α₁ while the largest is for λ.

7. COVID‐19 APPLICATION

How deadly is the coronavirus? In other words, how does mortality differ across countries? Actually, the true fatality rate is tricky to discover till now, but researchers are getting closer. Countries all over the world have reported very different case fatality ratios. When the total number of deaths from COVID‐19 are divided by the total number of cases not just the reported cases, the result is a statistic called the infection fatality rate (IFR), or colloquially, the death rate. Countries all over the world have reported very different case fatality ratios. Differences in fatality ratios or mortality numbers can be caused by: limiting testing and differences in the number of people who make a test, challenges in the attribution of the cause of death. Demographics: for example, older populations may have higher mortality. Moreover, healthcare system is an important factor on making these differences.

The centers for disease control and prevention currently changed the estimates of IFR between 0.26% and 0.65% and is still tricky. See Ref. ²²

For visualizing fatalities on COVID‐19 for different countries, we use the following data of the total number of cases per one million populations versus the total number of deaths per one million populations for different countries that have impacted number of infected cases and deaths as given in Ref. ²³ from January 22, 2020 to August 15, 2020 as shown in Table 4.

TABLE 4.

Total number of COVID‐19 cases versus the total deaths/1 M population

Country	USA	Brazil	India	Russia	South Africa	Mexico	Peru	Colombia	Spain	Iran
Total cases/1 M pop	16019	14631	1685	6150	9532	3815	14829	8057	7992	3937
Total deaths/1 M pop	506	485	33	104	181	418	651	265	611	224

Country	UK	Italy	France	Canada	Ecuador	Sweden	Panama	Belgium	Netherlands	Armenia
Total cases/1 M pop	4605	4156	3127	3187	5407	8225	17691	6469	3499	13696
Total deaths/1 M pop	686	583	465	238	337	571	389	852	359	271

Country	Kyrgyzstan	Bolivia	Switzerland	Moldova	Ireland	North Macedonia	Andorra	Channel Islands	Sent Maarten	Monaco
Total cases/1 M pop	6190	7983	4259	6998	5421	5800	12461	3442	4772	3514
Total deaths/1 M pop	226	322	230	213	359	254	673	270	396	102

The main descriptive statistics for the given data are summarized in Table 5 as follows.

TABLE 5.

Descriptive statistics for the data set

Statistics	Tot cases/1 M pop	Tot deaths/1 M pop
Minimum	1685	33
1st quartile	3906.5	229
Median	5975	348
Mean	7251.6	375.8
3rd quartile	8551.8	522.25
Maximum	17691	852
Coefficient of variation	0.5993	0.52606
Standard error	0.26598	0.737617
Pearson's correlation	0.281977
Kendall's tau	0.29667
Spearman's rho	0.444872
Rho/tau	1.499551

To find the initial estimates, we first test the goodness of fit for the marginal Rayleigh distribution by using Kolmogorov–Smirnov (KS) test and Anderson–Darling (AD), critical value ($Cv$). The results are summarized in Table 6 as follows, which gives a good fit for Rayleigh distribution.

TABLE 6.

Goodness‐of‐fit tests for the marginal to Rayleigh distribution

	MLE estimates	KS	p‐value	AD	Critical value at $\alpha =0.05$
Total cases/1 M pop	$\widehat{{\beta }_1}=5786$	$KS=0.13525$	0.59536	$AD=1.0603$	$CV=2.5018$
Total deaths/1 M pop	$\widehat{{\beta }_2}=299.85$	$KS=0.09803$	0.90842	$AD=0.24546$	$CV=2.5018$

Then via goodness‐of‐fit criteria, we make a comparison via different bivariate distributions models and BRPH model. The comparable bivariate distribution models are:

• (I)Bivariate Gumbel Rayleigh distribution

Gumbel–Barnett copula is due to Frees and Valdez, ²⁴ which is defined as

$$C\left(u,v\right)=u+v-1+\left(1-u\right)\left(1-v\right)\exp \left(-\theta Log\left(1-u\right)Log\left(1-v\right)\right)$$

where$0\le \theta \le 1$. Then, the Bivariate Gumbel Rayleigh ²⁵ can be defined as

$$g_{GR}\left(x_1,x_2;{\alpha }_1,{\alpha }_2,\alpha \right)=\frac{1}{4{\alpha }_1^4{\alpha }_2^4}{\mathrm{e}}^{-\frac{2x_2^2{\alpha }_1^2+x_1^2\left(x_2^2\theta +2{\alpha }_2^2\right)}{4{\alpha }_1^2{\alpha }_2^2}}{x}_1{x}_2\left(x_1^2\theta \left(x_2^2\theta +2{\alpha }_2^2\right)+2{\alpha }_1^2\left(x_2^2\theta -2\left(-1+\theta \right){\alpha }_2^2\right)\right)$$ (25)

where x _1, $x_2\ge 0,{\alpha }_1,{\alpha }_2>0,0\le \theta \le 1$.

• (II)Bivariate cubic Rayleigh distribution

In (2000), cubic copula has defined in Dolati et al. ¹² as

$$C\left(u,v\right)=uv\left(1+\theta \left(u-1\right)\left(v-1\right)\left(2u-1\right)\left(2v-1\right)\right)-1\le \theta \le 2$$

We used it to define bivariate cubic Rayleigh distribution as

$$\begin{array}{ccc}& & g_{CR}\left(x_1,x_2;{\alpha }_1,{\alpha }_2,\theta \right)\hfill \\ & & =\frac{{\mathrm{e}}^{-\frac{3}{2}\left(\frac{x_1^2}{{\alpha }_1^2}+\frac{x_2^2}{{\alpha }_2^2}\right)}{x}_1{x}_2\left(6\left(6-6{\mathrm{e}}^{\frac{x_1^2}{2{\alpha }_1^2}}+{\mathrm{e}}^{\frac{x_1^2}{{\alpha }_1^2}}\right)\theta -6{\mathrm{e}}^{\frac{x_2^2}{2{\alpha }_2^2}}\left(6-6{\mathrm{e}}^{\frac{x_1^2}{2{\alpha }_1^2}}+{\mathrm{e}}^{\frac{x_1^2}{{\alpha }_1^2}}\right)\theta +{\mathrm{e}}^{\frac{x_2^2}{{\alpha }_2^2}}\left(6\theta -6{\mathrm{e}}^{\frac{x_1^2}{2{\alpha }_1^2}}\theta +{\mathrm{e}}^{\frac{x_1^2}{{\alpha }_1^2}}\left(1+\theta \right)\right)\right)}{{\alpha }_1^2{\alpha }_2^2}\hfill \end{array}$$ (26)

where x _1, $x_2\ge 0,{\alpha }_1,{\alpha }_2>0,-1\le \theta \le 2$.

• (III)Bivariate Farlie–Gumbel Morgenstern Rayleigh distribution at ${\alpha }_1={\alpha }_2=\alpha$

Akhter and Hirari ¹³ have defined this distribution as

$$g_{FGMR1}\left(x_1,x_2;\alpha ,\theta \right)=\frac{{\mathrm{e}}^{-\frac{x_1^2+x_2^2}{{\alpha }^2}}{x}_1{x}_2\left(-2\left(-2+{\mathrm{e}}^{\frac{x_1^2}{2{\alpha }^2}}\right)\theta +{\mathrm{e}}^{\frac{x_2^2}{2{\alpha }^2}}\left(-2\theta +{\mathrm{e}}^{\frac{x_1^2}{2{\alpha }^2}}\left(1+\theta \right)\right)\right)}{{\alpha }^4}$$ (27)

where x _1, $x_2\ge 0,\alpha >0,-1\le \theta \le 1$.

• (IV)
  Bivariate Kumaraswamy type exponential distribution Mirhosseini et al. ⁹

  $$g_{BKE}\left(x_1,x_2;{\lambda }_1,{\lambda }_2,\alpha \right)=$$ (28)

  $${\lambda }_1{\lambda }_2\alpha e^{-\left({\lambda }_1{x}_1+{\lambda }_2{x}_2\right)}\left[1-\alpha \left(1-e^{-{\lambda }_1{x}_1}\right)\left(1-e^{-{\lambda }_2{x}_2}\right)\right]{\left[1-\left(1-e^{-{\lambda }_1{x}_1}\right)\left(1-e^{-{\lambda }_2{x}_2}\right)\right]}^{\alpha -2}$$

where, x _1, $x_2>0,{\lambda }_1,{\lambda }_2>0,0\le \alpha \le 1$

• (V)
  Bivariate Gamma on proportional hazard rate model is defined as

  $$g_{GPH}\left(x_1,x_2;{\alpha }_1,{\beta }_1,{\alpha }_2,{\beta }_2,\lambda \right)=\lambda \frac{{x_1}^{{\alpha }_1-1}}{{{\beta }_1}^{{\alpha }_1}\mathrm{\Gamma }\left({\alpha }_1\right)}{e}^{\frac{-x_1}{{\beta }_1}}\frac{{x_2}^{{\alpha }_2-1}}{{{\beta }_2}^{{\alpha }_2}\mathrm{\Gamma }\left({\alpha }_2\right)}{e}^{\frac{-x_2}{{\beta }_2}}{\left(1-\frac{{\mathrm{\Gamma }}_{\frac{x_1}{{\beta }_1}\left({\alpha }_1\right)}}{\mathrm{\Gamma }\left({\alpha }_1\right)}\frac{{\mathrm{\Gamma }}_{\frac{x_2}{{\beta }_2}\left({\alpha }_2\right)}}{\mathrm{\Gamma }\left({\alpha }_2\right)}\right)}^{\lambda -2}\left(1-\lambda \frac{{\mathrm{\Gamma }}_{\frac{x_1}{{\beta }_1}\left({\alpha }_1\right)}}{\mathrm{\Gamma }\left({\alpha }_1\right)}\frac{{\mathrm{\Gamma }}_{\frac{x_2}{{\beta }_2}\left({\alpha }_2\right)}}{\mathrm{\Gamma }\left({\alpha }_2\right)}\right)$$ (29)

where, x _1, $x_2>0,{\alpha }_1,{\beta }_1,{\alpha }_2,{\beta }_2>0,0\le \lambda \le 1$.

Using the initial estimates, the MLEs are obtained of the parameters. In order to investigate the potentiality of fitting for the proposed model $BRPH$ than the comparable Bivariate Rayleigh models shown above, we will use the Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Hannan–Quinn Information Criterion (HQIC), and Consistent AIC (CAIC). The obtained results indicate that BRPH model provides a better fit to the data set since it has lowest values for the goodness‐of‐fit criteria. The results are summarized in Table 7.

TABLE 7.

Goodness‐of‐fit criteria for BRPH and other models

Distribution	MLEs	Log	AIC	BIC	CAIC	HQIC
$BRPH$	$\widehat{{\alpha }_1}=272.542$ $\widehat{{\alpha }_2}=5539.63$ $\widehat{\lambda }=0.841509$	−488.562	983.124	987.328	984.047	984.469
Gumbel Rayleigh	$\widehat{{\alpha }_1}=312.218,$ $\widehat{{\alpha }_2}=6270.33$ at $\widehat{\alpha }=0.3$	−492.297	988.596	991.398	989.04	989.492
Cubic Rayleigh	$\widehat{{\alpha }_1}=297.567,$ $\widehat{{\alpha }_2}=5940.56$ $\widehat{\theta }=0.117251$	−489.028	984.979	988.26	984.979	985.401
One parameter Morgenstern Rayleigh	$\widehat{\alpha }=3807.38$ $\widehat{\theta }=-0.98$	−620.256	1244.51	1247.32	1244.96	1245.41
Bivariate Kumaraswamy type exponential	$\widehat{{\alpha }_1}=0.00646$ $\widehat{{\alpha }_2}=0.00032$ $\widehat{\lambda }=0.393856$	−500.004	1006.01	1010.21	1006.93	1007.35
Bivariate Gamma proportional hazard	$\widehat{{\alpha }_1}=3.24365,$ $\widehat{{\alpha }_2}=1.868$ $\widehat{{\beta }_1}=129.17,$ $\widehat{{\beta }_2}=2724.22,$ $\widehat{\alpha }=1$	−489.422	988.844	995.85	991.344	991.085

8. CONCLUSION

In this paper, the BRPH distribution is proposed and applied on COVID‐19 data. This distribution has been constructed with the proportional hazard model. Its joint PDF and CDF has been expressed in a simple form, which makes BRPH used definitely in practice. The statistical and Reliability characteristics have been studied. The related copula has been obtained which is one of the Archimedean family and the dependence properties are considered. The point and interval parameter estimation are discussed via different methods. Simulation study is obtained to compare the performance of the parameter estimates via the different estimation approaches. The study showed that MLE approach mostly provides better results for the three parameters. Some constructed models of bivariate Rayleigh distribution have been built via copula concepts and others used for comparison in a practical application of COVID‐19. The study verified that BRPH could provide a flexible good‐fitted model compared with other competitive bivariate distribution models which describe the total number of infected cases/1 M population versus the total number of deaths per 1 M population for different countries, which have notable number of deaths due to coronavirus. Then, BRPH accommodate one of the good probabilistic model, which can model the burden of fatality of COVID‐19. For future research, a multivariate Rayleigh Proportional Hazard distribution can be constructed with related copula.

Biographies

Wafaa Anwar Abd El‐Latif Hassanein, Associate Professor of Mathematical Statistics, Mathematics Department, Faculty of Science, Tanta University, Tanta, Egypt. She got her B.Sc., 2000, by excellence with honor degree, she got M.Sc. degree, 2004, in Lerch Family of distributions, and Ph.D. degree, 2007, titled “Uncertainty in Statistics,” Faculty of Science, Tanta University. She is a member of ERS Group and the Egyptian Mathematical Society. Her main research interests are distribution theory, statistical inference, and optimal design of experiments.

Marwa Moghazy Attia Seyam, Lecturer of Mathematical Statistics, Mathematics Department, Faculty of Science, Tanta University, Tanta, Egypt, and Associate Professor of Mathematics Department, Faculty of Science and Arts, Jouf University, Sakaka, Saudi Arabia. She got her B.Sc., 2006, by excellence with honor degree, she got M.Sc. degree, 2009, In Optimal Design, and a Ph.D. degree, 2013, in nonparametric models, Faculty of Science, Tanta University. She is a member of the Egyptian Mathematical Society. Her main research interests are distribution theory, statistical inference, and optimal design of experiments.

THE OBSERVED FISHER INFORMATION MATRIX

$$\begin{array}{ccc}& & {\displaystyle \frac{{\partial }^2\log L}{\partial {\alpha }_1^2}}=\left(\lambda -2\right)\hfill \\ & & \sum _{i=1}^n\left[-\frac{{\mathrm{e}}^{-\frac{x_{1i}^2}{{\alpha }_1^2}}{\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)}^2{X}_{1i}^4}{{\left(1-\left(1-{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\right)\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)\right)}^2{\alpha }_1^6}+\frac{{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)X_{1i}^4}{\left(1-\left(1-{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\right)\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)\right){\alpha }_1^6}-\frac{3{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)X_{1i}^2}{\left(1-\left(1-{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\right)\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)\right){\alpha }_1^4}\right]\hfill \\ & & +\sum _{i=1}^n\left(-\frac{{\mathrm{e}}^{-\frac{x_{1i}^2}{{\alpha }_1^2}}{\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)}^2{\lambda }^2{x}_{1i}^4}{{\left(1-\left(1-{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\right)\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)\lambda \right)}^2{\alpha }_1^6}+\frac{{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)\lambda x_{1i}^4}{\left(1-\left(1-{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\right)\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)\lambda \right){\alpha }_1^6}-\frac{3{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)\lambda x_{1i}^2}{\left(1-\left(1-{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\right)\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)\lambda \right){\alpha }_1^4}\right)\hfill \\ & & +\sum _{i=1}^n\left(-\frac{3x_{1i}^2}{{\alpha }_1^4}+\frac{2}{{\alpha }_1^2}\right)\hfill \end{array}$$

$$\frac{{\partial }^2\log L}{\partial {\lambda }^2}=-\frac{n}{{\lambda }^2}+\sum _{i=1}^n-\frac{{\left(1-{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\right)}^2{\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)}^2}{{\left(1-\left(1-{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\right)\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)\lambda \right)}^2}$$

$$\begin{array}{ccc}& & \frac{{\partial }^2\log L}{\partial {\alpha }_1\partial {\alpha }_2}=\left(\lambda -2\right)\sum _{i=1}^n\left[-\frac{{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}-\frac{x}{2{\alpha }_2^2}}\left(1-{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\right)\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)x_{1i}^2{x}_{2i}^2}{{\left(1-\left(1-{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\right)\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)\right)}^2{\alpha }_1^3{\alpha }_2^3}-\frac{{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}-\frac{x}{2{\alpha }_2^2}}{x}_{1i}^2{x}_{2i}^2}{\left(1-\left(1-{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\right)\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)\right){\alpha }_1^3{\alpha }_2^3}\right]\hfill \\ & & +\sum _{i=1}^n\left[-\frac{{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}-\frac{x_{2i}^2}{2{\alpha }_2^2}}\left(1-{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\right)\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right){\lambda }^2{x}_{1i}^2{x}_{2i}^2}{{\left(1-\left(1-{\mathrm{e}}^{-\frac{x}{2{\alpha }_1^2}}\right)\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)\lambda \right)}^2{\alpha }_1^3{\alpha }_2^3}-\frac{{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}-\frac{x}{2{\alpha }_2^2}}\lambda x_{1i}^2{x}_{2i}^2}{\left(1-\left(1-{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\right)\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)\lambda \right){\alpha }_1^3{\alpha }_2^3}\right]\hfill \end{array}$$

$$\begin{array}{ccc}& & \frac{{\partial }^2\log L}{\partial {\alpha }_2\partial \lambda }=\sum _{i=1}^n\left(\frac{{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}{\left(1-{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\right)}^2\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)\lambda X_{2i}^2}{{\left(1-\left(1-{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\right)\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)\lambda \right)}^2{\alpha }_2^3}+\frac{{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\left(1-{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\right)x_{2i}^2}{\left(1-\left(1-{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\right)\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)\lambda \right){\alpha }_2^3}\right)\hfill \\ & & +\sum _{i=1}^n\frac{{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\left(1-{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\right)X_{2i}^2}{\left(1-\left(1-{\mathrm{e}}^{-\frac{x_{1i}^2}{2{\alpha }_1^2}}\right)\left(1-{\mathrm{e}}^{-\frac{x_{2i}^2}{2{\alpha }_2^2}}\right)\right){\alpha }_2^3}\hfill \end{array}$$

Hassanein WAAE‐L, Seyam MMA. Structure of bivariate Rayleigh proportional hazard rate model with its associated copula applied on COVID‐19 data. Qual Reliab Eng Int. 2022;38:3451–3469. 10.1002/qre.3143

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are openly available in “Reported Cases and Deaths by Country or Territory” at https://www.worldometers.info/coronavirus/. ²³