Necessary and sufficient conditions for asymptotic normality of trimmed L-statistics

Abstract

Necessary and sufficient conditions are obtained for the asymptotic normality of L-statistics when the weights are formed by means of a general class of smooth score functions J. This asymptotic normality problem is solved for trimming fixed fractions, vanishing fractions and fixed numbers of observations. In the process of obtaining the solution, all possible limit laws of such statistics are described; these include a generalization of the class of infinitely divisible laws that may be of independent interest. A further set of sufficient conditions is derived for the asymptotic normality of L-statistics that rival, in terms of mildness of regularity, those conditions required by all previous theorems on the subject. Asymptotic normality for a generalized trimmed mean form of student's t-statistic is also included. It is asymptotically normal whenever the observations are in the domain of attraction of any stable law.