Cramér-type moderate deviations for intermediate trimmed means

Abstract

ABSTRACTIn this article, we establish Cramér-type moderate deviation results for (intermediate) trimmed means Tn = n− 1∑n − mni = kn + 1Xi: n, where Xi: n's are the order statistics corresponding to the first n observations in a sequence X1, X2, … of independent identically distributed random variables with F. We consider two cases of intermediate and heavy trimming. In the former case, when max (αn, βn) → 0 (αn = kn/n, βn = mn/n) and min (kn, mn) → ∞ as n → ∞, we obtain our results under a natural moment assumption and a mild condition on the rate at which αn and βn 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 αn, 1 − βn.