Second Order Approximations for Slightly Trimmed Sums

Details Second Order Approximations for Slightly Trimmed Sums Second Order Approximations for Slightly Trimmed Sums

2011-04-17

arxiv.org/abs/1104.3347v1

Mathematics

Probability

37 pages, to appear in Theory Probab. Appl

Scientific paper

We investigate the second order asymptotic behavior of trimmed sums $T_n=\frac 1n \sum_{i=\kn+1}^{n-\mn}\xin$, where $\kn$, $\mn$ are sequences of integers, $0\le \kn < n-\mn \le n$, such that $\min(\kn, \mn) \to \infty$, as $\nty$, the $\xin$'s denote the order statistics corresponding to a sample $X_1,...,X_n$ of $n$ i.i.d. random variables. In particular, we focus on the case of slightly trimmed sums with vanishing trimming percentages, i.e. we assume that $\max(\kn,\mn)/n\to 0$, as $\nty$, and heavy tailed distribution $F$, i.e. the common distribution of the observations $F$ is supposed to have an infinite variance. We derive optimal bounds of Berry -- Esseen type of the order $O\bigl(r_n^{-1/2}\bigr)$, $r_n=\min(\kn,\mn)$, for the normal approximation to $T_n$ and, in addition, establish one-term expansions of the Edgeworth type for slightly trimmed sums and their studentized versions. Our results supplement previous work on first order approximations for slightly trimmed sums by Csorgo, Haeusler and Mason (1988) and on second order approximations for (Studentized) trimmed sums with fixed trimming percentages by Gribkova and Helmers (2006, 2007).

Affiliated with

Also associated with

No associations

Rating

Second Order Approximations for Slightly Trimmed Sums does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Second Order Approximations for Slightly Trimmed Sums, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Second Order Approximations for Slightly Trimmed Sums will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-71106