Dependence properties of bivariate distributions with proportional (reversed) hazards marginals

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

“…Let (𝑋 1 , 𝑋 2 ) be a bivariate Rayleigh random vector with scale parameters 𝛼 1 , 𝛼 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 (𝑥 1 , 𝑥 2 ; 𝛼 1 , 𝛼 2 , 𝜆) = 𝜆…”

“…Due to Dolati et al, 12 the following proposition is summarized Proposition 4: Let 𝐶 𝜆 (𝑢, 𝑣) be a copula defined in Equation (15) then for every 0 < 𝜆 ≤ 1…”

“…In (2000), cubic copula has defined in Dolati et al 12 . as $$\begin{equation*}C\;\left( {u,v} \right) = \;uv\;\left( {1 + \theta \left( {u - 1} \right)\left( {v - 1} \right)\left( {2u - 1} \right)\left( {2v - 1} \right)} \right)\;\qquad - 1 \le \theta \le 2\end{equation*}$$…”

“…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.…”

A generalized bivariate lifetime distribution based on parallel-series structures

“…Proposition 3.1. Suppose that (T 1 , T 2 ) have the bivariate distribution function as given in (4), then…”