Cram\'er type moderate deviations for intermediate trimmed means

TL;DR

This paper derives Cramér type moderate deviation results for intermediate and heavy trimmed means of i.i.d. data, under mild moment and smoothness conditions, extending the understanding of their tail behavior.

Contribution

It provides the first Cramér type moderate deviation results for trimmed means in both intermediate and heavy trimming scenarios, with minimal assumptions.

Findings

Results hold under natural moment conditions for intermediate trimming.

No moment conditions needed for heavy trimming, only smoothness of inverse distribution.

Applicable to a wide class of distributions with mild regularity assumptions.

Abstract

In this article we establish Cram\'er 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 $\alpha_n$, $1-\beta_n$.