TY - JOUR

T1 - Second Order Approximations for Slightly Trimmed Means

AU - Gribkova, N.V.

AU - Helmers, R.

N1 - N.V.Gribkova, R.Helmers Second Order Approximations for Slightly Trimmed Means. Theory of Probability and its Applications, 2014. — Vol. 58, — № 3. — P. 383–412

PY - 2014

Y1 - 2014

N2 - We investigate the second order asymptotic behavior of distributions of statistics $T_n=\frac 1n \sum_{i=\kn+1}^{n-\mn}\xin$, where $\kn$, $\mn$ are sequences of integers, $0\le \kn < n-\mn \le n$, and $r_n:=\min(\kn, \mn) \to \infty$, as $\nty$, the $\xin$'s denote the order statistics corresponding to a sample $X_1,\dots,X_n$ of $n$ i.i.d. r.v.'s. In particular, we focus on the case of slightly trimmed means 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 -- Ess\'{e}en type of the order $O\bigl(r_n^{-1/2}\bigr)$ for the normal approximation to $T_n$ and, in addition, establish one-term expansions of the Edgeworth type for slightly trimmed means and their studentized versions. Our results supplement previous work on first order approximations for slightly trimmed sums by Csorgo, Haeusler \& Mason (1988) and on second order approximations for (Studentized) trimmed means with fixed trimming percentages by Gribkova \& Helmers~(2006, 2007).

AB - We investigate the second order asymptotic behavior of distributions of statistics $T_n=\frac 1n \sum_{i=\kn+1}^{n-\mn}\xin$, where $\kn$, $\mn$ are sequences of integers, $0\le \kn < n-\mn \le n$, and $r_n:=\min(\kn, \mn) \to \infty$, as $\nty$, the $\xin$'s denote the order statistics corresponding to a sample $X_1,\dots,X_n$ of $n$ i.i.d. r.v.'s. In particular, we focus on the case of slightly trimmed means 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 -- Ess\'{e}en type of the order $O\bigl(r_n^{-1/2}\bigr)$ for the normal approximation to $T_n$ and, in addition, establish one-term expansions of the Edgeworth type for slightly trimmed means and their studentized versions. Our results supplement previous work on first order approximations for slightly trimmed sums by Csorgo, Haeusler \& Mason (1988) and on second order approximations for (Studentized) trimmed means with fixed trimming percentages by Gribkova \& Helmers~(2006, 2007).

KW - slightly trimmed mean

KW - intermediate sample quantiles

KW - asymptotic normality

KW - Berry--Esseen bound

KW - Edgeworth expansion

U2 - 10.1137/S0040585X97986618

DO - 10.1137/S0040585X97986618

M3 - Article

VL - 58

SP - 383

EP - 412

JO - Theory of Probability and its Applications

JF - Theory of Probability and its Applications

SN - 0040-585X

IS - 3

ER -