Cramér type moderate deviations for intermediate trimmed means

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 $\max(α_n,β_n)\to 0$ ($α_n=k_n/n$, $β_n=m_n/n$) 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 $α_n$ 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 $α_n$, $1-β_n$.

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