Limiting Multivariate Distributions of Intermediate Order Statistics

Abstract

Let $Z^{(n)}_m$ represent the $m$th largest order statistic in a random sample of size $n$ from a distribution $F$. If $m = m(n)$ is an intermediate sequence such that $m \rightarrow \infty$ and $m/n \rightarrow 0$ as $n \rightarrow \infty$, the intermediate order statistics of the form $Z^{(n)}_{\lbrack mt_1\rbrack}, \cdots, Z^{(n)}_{\lbrack mt_k\rbrack}$, for $0 < t_1 < \cdots < t_k$, can be used jointly for making statistical inferences about the upper tail of $F$. We find the asymptotic joint distribution of order statistics of this form, for various types of underlying distributions $F$, by determining the limit (weak convergence) of a stochastic process of the form $(Z^{(n)}_{\lbrack mt\rbrack} - \beta^{(n)}_{mt})/\alpha^{(n)}_{mt}, t > 0$, for appropriate normalizing functions $\alpha^{(n)}_{mt}, > 0, \beta^{(n)}_{mt}$.