Cramér type moderate deviations for intermediate trimmed means

Abstract

In this article we establish Cramér type moderate deviation results for (intermediate) trimmed means $T_n=n^{-1}{\sum }_{i=k_n+1}^{n-m_n}{X}_{i:n}$, where $X_{i:n}$ -- the order statistics corresponding to the first $n$ observations of a~sequence $X_1,X_2,\dots$ of i.i.d random variables with $df$ $F$. We consider two cases of intermediate and heavy trimming. In the former case, when $\max ({\alpha }_n,{\beta }_n)\to 0$ (${\alpha }_n=k_n/n$, ${\beta }_n=m_n/n$) and $\min (k_n,m_n)\to \infty$ as $n\to \infty$, we obtain our results under a~natural moment condition and a~mild condition on the rate at which ${\alpha }_n$ and ${\beta }_n$ 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 ${\alpha }_n$, $1-{\beta }_n$.