Stochastic comparison of parallel systems with heterogeneous dependent exponential components

Let $X=(X_1,\dots ,X_n)$ and $Y=(Y_1,\dots ,Y_n)$ be two random vectors with common Archimedean copula with generator function $\varphi$, where, for $i=1,\dots ,n$, $X_i$ is an exponential random variable with hazard rate ${\lambda }_i$ and $Y_i$ is an exponential random variable with hazard rate $\lambda$. In this paper we prove that under some sufficient conditions on the function $\varphi$, the largest order statistic corresponding to $X$ is larger than that of $Y$ according to the dispersive ordering and hazard rate ordering. The new results generalized the results in Dykstra et al. (1997) and Khaledi and Kochar (2000). We show that the new results can be applied to some well known Archimedean copulas.