ACTA issues

Limit theorems for linear combinations of extreme values with applications to inference about the tail of a distribution

László Viharos

Acta Sci. Math. (Szeged) 60:3-4(1995), 761-778

Abstract. Let $X_{1,n}\le\cdots \le X_{n,n}$ be the order statistics of $n$ independent random variables with a common distribution function $F$ belonging to the domain of attraction of an extreme value distribution and let $k_n$ be positive integers such that $k_n\to\infty $ and $k_n/n\to\alpha $ as $n\to\infty $, where $0\le\alpha < 1$. Given known constants $d_{i,n}$, $1\le i\le n$, that are all specified by the statistician, consider the linear combination $T_n(k,k_n)=\sum_{i=k+1}^{k_n} d_{n+1-i,n}f(X_{n+1-i,n})$ of extreme values, where $k\ge0$ is any fixed integer and $f$ is a Borel-measurable function. With suitable $f$ and normalizing and centering constants $A_n>0$ and $C_n$ we determine the limiting distribution of the sequence $(T_n(k,k_n)-C_n)/A_n.$ Using linear combinations of extreme values, we also generalize the classical Hill estimator for the index of a distribution function $F$ with a regularly varying upper tail.

AMS Subject Classification (1991): 62E20; 62G05, 62G30

Received June 21, 1994, and in revised form October 15, 1994. (Registered under 5664/2009.)