Probability Inequalities for Generalized L-Statistics

Abstract

where Xn:1 ≤ · · · ≤ Xn:n are the order statistics based on the sample {Xi; i ≤ n} and hni : R → R, i = 1, . . . , n, are measurable functions. In particular, if hni(y) = cnih(y) and h(y) is monotone then Φn represents the classical L-statistics. Functionals (1) in this general form are called generalized L-statistics. For the first time, these statistics were introduced in [1, 2] where asymptotic expansions for the distributions of these statistics were given in some particular cases. The Fourier analysis of the distributions of Φn is contained in [3]. Note that the integral-type statistics (integral functionals of the empirical distribution function, for example, the Anderson–Darling–Cramer statistics) can be represented as (1), but not as the classical L-statistics (see e.g. [1, 3]). The main purpose of this paper is to obtain upper bounds for the tail probability and moments of Φn. Exponential bounds for the tail probabilities of the classical L-statistics were obtained in [4] by means of approximation of L-statistics by U -statistics with nondegenerate kernels, which makes it possible to reduce the problem to analogous problems for sums of independent real-valued random variables. The approach of the present paper illustrates the capabilities of multivariate analysis: the problems in question are reduced to analogous problems for sums of independent random elements taking values in a functional Banach space. In the previous paper [5], containing some moment inequalities for generalized L-statistics, we suggested an analogous approach using a special property of the order statistics based on a sample from an exponential distribution. In the present paper, to study generalized L-statistics we essentially use the properties of order statistics based on a sample from the (0, 1)-uniform distribution, although we impose no additional restrictions on the sample distribution for the so-called L-statistics with separated kernels which are introduced below. Note also that the term “generalized L-statistics” was introduced in [6] where a generalization of the classical L-statistics theory was considered in a somewhat different aspect related to another construction of order statistics.