A NOTE ON GENERALIZED PARETO DISTRIBUTIONS AND THE K UPPER EXTREMES

Denote by D(n, k) the maximum distance between the distribution function of the kth-largest order statistic in an iid sample of size n, equally standardized in k, and of its corresponding limiting extreme value distribution. Suppose that the underlying distribution function is ultimately continuous and strictly increasing. It is shown in the present paper that D(n,k) converges to zero for any sequence k=k (n) with {Mathematical expression} if and only if the underlying distribution is ultimately a generalized Pareto distribution. Thus gPds do not only yield the best rate of joint convergence of extremes, but they are also the only distributions where there actually is convergence. © 1990 Springer-Verlag.