Approximate moments of extremes

Abstract

Let $M_{n, i}$ be the ith largest of a random sample of size n from a cumulative distribution function F on $\mathbb{R} = (-\infty, \infty)$ . Fix $r \geq1$ and let $\mathbf{M}_{n} = ( M_{n, 1}, \ldots, M_{n, r} )^{\prime}$ . If there exist $b_{n}$ and $c_{n} > 0$ such that as $n \rightarrow \infty$ , $( M_{n, 1} - b_{n} ) / c_{n} \stackrel{\mathcal {L}}{\rightarrow} Y_{1} \sim G$ say, a non-degenerate distribution, then as $n \rightarrow\infty$ , $\mathbf{Y}_{n} = ( \mathbf{M}_{n} - b_{n} {\mathbf{1}}_{r} ) / c_{n} \stackrel{\mathcal{L}}{\rightarrow} \mathbf{Y}$ , where for $Z_{i} = -\log G ( Y_{i} )$ , $\mathbf{Z} = ( Z_{1}, \ldots, Z_{r} )^{\prime}$ has joint probability density function $\exp ( -z_{r} )$ on $0 < z_{1} < \cdots< z_{r} < \infty$ and ${\mathbf{1}}_{r}$ is the r-vector of ones. The moments of Y are given for the three possible forms of G. First approximations for the moments of $\mathbf{M}_{n}$ are obtained when these exist.