Article

Title: Expansions for the Joint Distribution of the Sample Maximum and Sample Estimate

Issue: Volume 70 Series A Part 1 Year 2008
Abstract
Let $F_n$ be the empirical distribution of a random sample in $R^p$ from a distribution $F$. Let $M_n$ be the componentwise sample maximum and $T(F)$ a smooth functional in $R^q$. Let $\hat{\theta} = T(F_n)$. We use the conditional Edgeworth expansion for $\hat{\theta} | (M_n \le y)$ to obtain expansions for the joint distribution of $(\hat{\theta}, M_n)$. For $T(F) = \mu$ and $\mu_2$, their degree of dependence as measured by the strong-mixing coefficient $\alpha(\hat{\theta}, M_n)$ is shown to be $O(n^{-1/2})$ for a class of distributions associated with the EV3 (Weibull), $O(n^{-1/2} \log^{i\nu} n)$ for two classes associated with the EV1 (Gumbel) and $O(n^{i/\theta - 1/2})$ for a class associated with the EV2 ($\theta$) ) (Frechet), where $i$ is the degree of $T(F)$, that is $i = 1$ for $\mu$ and $i = 2$ for $\mu_2$, $\nu = 1$ for a class that includes the gamma, and $\nu = 1/2$ for a class that includes the normal.