Generalized L-, M-, and R-statistics. (English) Zbl 0538.62015

In this paper a new class of statistics, called generalized L-statistics (GL-statistics for short) is introduced and studied. The class of GL- statistics is quite large. It includes U-statistics, linear functions of order statistics and many other statistics of interest, such as ”trimmed U-statistics” and the Hodges-Lehmann estimator of location. GL-statistics can be viewed as L-functionals of the empirical distribution function of U-statistic structures. This representation is employed by the author to prove, with the aid of the differential statistical function approach, asymptotic normality for generalized L-statistics. A related CLT for GL- statistics was obtained by B. W. Silverman, Ann. Probab. 11, 745- 751 (1983; Zbl 0514.60040), using a different method of proof.
Similar generalizations of M- and R-statistics are briefly sketched. Recent further as yet unpublished work by S. Csörgö, L. Horvath and the author (A strong approximation for the empirical process of U-statistic structure) and by P. Janssen, the author and the reviewer (Glivenko-Cantelli theorems for empirical distribution function of U-statistic structure, and a Berry-Esséen theorem for GL-statistics) is also mentioned.
Reviewer: R.Helmers


62E20 Asymptotic distribution theory in statistics
60F15 Strong limit theorems
62G30 Order statistics; empirical distribution functions


Zbl 0514.60040
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