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