Randomly trimmed l-statistics

Abstract

Central limit theorems (CLT) and strong laws of large numbers (SLLN) are obtained for randomly trimmed L-statistics. When the fractions trimmed converge to a and 1−b with 0<a<b<1 there are no restrictions on the df's of the rv's, except for necessary continuity of F−1 at a and b; but the limiting rv has several contributing terms, making studentization complicated unless the random trimming fractions converge fast enough. When a=0 andb=1 the natural restriction on the rate of convergence of the trimming fractions that allows the ‘right’ conclusion is more severe. But, this is a most reasonable way to trim. The technique is honed enough to yield the classical CLT and SLLN as immediate corollaries, even with random trimming. This is the way data analysts should, and do, trim; they have tacitly assumed such a CLT and LLN for years.