ON THE LIMIT BEHAVIOUR OF THE JOINT DISTRIBUTION FUNCTION OF ORDER STATISTICS

H. FINNER AND M. ROTERS

FB IV-Mathematik/Statistik, Universität Trier, D-54286 Trier, Germany

(Received December 11, 1992; revised June 11, 1993)

Abstract. For k ∈ _{0} fixed we consider the joint distribution function F_{n}^{k} of the n − k smallest order statistics of n real-valued independent, identically distributed random variables with arbitrary cumulative distribution function F. The main result of the paper is a complete characterization of the limit behaviour of F_{n}^{k}(x_{1}, …, x_{n−k}) in terms of the limit behaviour of n(1−F (x_{n})) if n tends to infinity, i.e., in terms of the limit superior, the limit inferior, and the limit if the latter exists. This characterization can be reformulated equivalently in terms of the limit behaviour of the cumulative distribution function of the (k + 1)-th largest order statistic. All these results do not require any further knowledge about the underlying distribution function F.

Key words and phrases: Extreme order statistic, extreme value distribution, joint distribution function of order statistics, multiple comparisons, Poisson distribution, uniform distribution.