Cramér-type moderate deviations for intermediate trimmed means

Abstract

In this article we establish Cramér type moderate deviation results for (intermediate) trimmed means T n = n −1 n−mn i=kn+1 X i:n , where X i:n -the order statistics corresponding to the first n observations of a sequence X 1 , X 2 , . . . of i.i.d random variables with df F . We consider two cases of intermediate and heavy trimming. In the former case, when max(α n , β n ) → 0 (α n = k n /n, β n = m n /n) and min(k n , m n ) → ∞ as n → ∞, we obtain our results under a natural moment condition and a mild condition on the rate at which α n and β n tend to zero. In the latter case we do not impose any moment conditions on F ; instead, we require some smoothness of F −1 in an open set containing the limit points of the trimming sequences α n , 1 − β n .