Dependent Stress-Strength Reliability Model

Dependent Stress-Strength Reliability Model

For a $2$-dimensional continuous random vector $(X,Y)$ with joint cumulative distribution function (CDF) $H(x,y)$ and univariate marginal CDFs $F(x)$ and $G(y)$. Then, based on Sklar’s Theorem there exist a unique $2$-dimensional copula function $C:[0,1]^2 \rightarrow [0,1]$ satisfying $$H(x,y)=C \big( F(x), G(y) \big).$$ Let $c$ and $h$ be the corresponding joint probability density function (PDF) of $C$ and $H$, respectively, and $f,g$ are the corresponding PDF of $F, G$, we have $$h(x,y) = \frac{\partial H(x,y)}{\partial x \partial y} = \frac{\partial C(F(x), G(y))}{\partial x \partial y} = c \big( F(x), G(y) \big) f(x) g(y).$$

Let the strength $X$ and stress $Y$ variables be dependent, with their dependence modeled via a two-dimensional copula function $C(u,v)$ and joint PDF $h(x,y)$. Then, the dependent stress–strength reliability $R$ is given by $$\begin{eqnarray*} R = P(X>Y) &=& \int_{0}^{\infty} \int_{0}^{x} h(x,y) \mathrm{d}y \mathrm{d}x = \int_{0}^{\infty} \int_{0}^{x} \frac{\partial C^2(u,v)}{\partial u \partial v} \Biggr|_{ \begin{smallmatrix} u=F(x) \\ v = G(y) \end{smallmatrix} } f(x) g(y) \mathrm{d}y \mathrm{d}x \\ &=& \int_{0}^{\infty} \frac{\partial C(u,v)}{\partial u} \Biggr|_{ \begin{smallmatrix} u=F(x) \\ v = G(x) \end{smallmatrix} } f_X(x) \mathrm{d}x. \end{eqnarray*}$$

The joint distribution function of the two-dimensional Clayton copula, along with its joint probability density function, are given by $$C_{\theta}(u,v) = (u^{-\theta} + v^{-\theta} - 1)^{-1/\theta},$$ and $$c_{\theta}(u,v) = (\theta +1) u^{-(\theta + 1)} v^{-(\theta + 1)} \big( u^{-\theta} + v^{-\theta}-1 \big)^{-\left (\frac{1}{\theta} + 2 \right)},$$ where $\theta >0$.

When $X \sim MWD(a_1,b_1,\lambda_1)$, $Y \sim MWD(a_2,b_2,\lambda_2)$ with the two-dimensional Clayton copula from, $R$ becomes $$\begin{eqnarray*} R &=& \int_{0}^{\infty} F_X(x)^{-(\theta + 1)} \big(F_X(x)^{-\theta} + G_Y(x)^{-\theta}-1 \big) ^{-\left (\frac{1}{\theta}+1 \right)} f_X(x) \mathrm{d}x \\ &=& \int_{0}^{1} t^{-(\theta +1)} \big( t^{-\theta} + G_Y(F_{X}^{-1}(t))^{-\theta} -1 \big)^ {-\left(\frac{1}{\theta}+1 \right) } \mathrm{d}t, \end{eqnarray*}$$ where $$F_X(x) \equiv F_X(x;a_1,b_1,\lambda_1) = 1- \exp(-a_1 x^{b_1} e^{\lambda_1 x}),$$ and $$G_Y(y) \equiv G_Y(y;a_2,b_2,\lambda_2) = 1- \exp(-a_2 y^{b_2} e^{\lambda_2 y}).$$

Further details can be found in Kızılaslan (2026).

References

Kızılaslan, Fatih. (2026). Reliability estimation in dependent stress–strength model with Clayton copula and modified Weibull margins.arXiv:2604.12130