Let $\bm X=(X_1,\ldots,X_n)$ and $\bm Y=(Y_1,\ldots,Y_n)$ be two random vectors with common Archimedean copula with generator function $\phi$, where, for $i=1,\ldots,n$, $X_i$ 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 $\phi$, the largest order statistic corresponding to $\bm X$ is larger than that of $\bm Y$ according to the dispersive ordering and hazard rate ordering. The new results generalized the results in Dykstra, Kochar, and Rojo [{\it Journal of Statistical planning and Inference} {\bf 65} (1997) 203-211] and Khaledi and Kochar [{\it Journal of Applied Probability} {\bf 37} (2000) 283-291]. We show that the new results can be applied to some well known Archimedean copulas.
