On Analogues of Berry–Esseen Inequality for Truncated Linear Combinations of Order Statistics

Gribkova, Nadezhda

doi:10.1137/1138011

Cited by 7 publications

(5 citation statements)

References 4 publications

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“…[11,10]). Therefore, in absence of any moment assumptions on the distribution F , we can replace with impunity T N by T N (which has finite moments of arbitrary order) when proving Theorems 2.1 and 2.3.…”

Section: Lemma 44 Suppose That the Conditions Of Lemma 42 Hold Trumentioning

confidence: 99%

On the Edgeworth expansion and the M out of N bootstrap accuracy for a Studentized trimmed mean

Gribkova

Helmers²

2007

Math. Meth. Stat.

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Abstract-We investigate the second order accuracy of the M out of N bootstrap for a Studentized trimmed mean using the Edgeworth expansion derived in a previous paper. Some simulations, which support our theoretical results, are also given. The effect of extrapolation in conjunction with the M out of N bootstrap for Studentized trimmed means is briefly discussed. As an auxiliary result we obtain a Bahadur's type representation for an M out of N bootstrap quantile. 1. INTRODUCTION This article is closely connected with our previous paper [11], where the validity of the one-term Edgeworth expansion (EE) and the empirical Edgeworth expansion (EEE) for a Studentized trimmed mean was established and explicit formulas for the leading terms of the EE were obtained. We study two second-order approximations to the distribution function (df ) of a Studentized trimmed mean: EE and bootstrap.During the past twenty five years the attention of many authors was focussed on Efron's bootstrap ([8]), and nowadays there exists a voluminous literature on this topic. The consistency of the standard nonparametric, or naive, bootstrap was proved for many interesting statistics, at least for the asymptotically normal ones (see [1,4,6,7,12,13,15,25] and references therein). One of the main reasons of interest in the bootstrap and its application in statistics is the second order accuracy property: under proper conditions the bootstrap approximation to the distribution function (df ) of a pivotal statistic is more accurate than the normal one. This beneficial property of the bootstrap was proved for the sample mean [25], for the class of Hoeffding's U -statistics (cf. [15]) and for some other statistics. The usual way to prove this fact is based on the congruence of the one-term EE for the pivotal statistic (in the 'real world') and the EE for its bootstrap counterpart (in the 'bootstrap world'). Typically the structure of the one-term EE for the bootstrapped statistic is the same as the one for the pivotal statistic, when the parameters of the formula of the first leading term of EE are replaced by their empirical counterparts (plug-in estimators). So, the application of a relevant version of the Law of Large Numbers implies the second order accuracy of the bootstrap (cf. [12,15,25]). However, the case of the trimmed mean is a special one. The problem is connected with the difficulty in obtaining the explicit formula for the oneterm EE (cf. [13]).In this paper we establish the validity of a one-term EE for the bootstrapped Studentized trimmed mean (in the bootstrap world). We also obtain an explicit formula for the M −1/2 -term (correcting for

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Section: Lemma 44 Suppose That the Conditions Of Lemma 42 Hold Trumentioning

confidence: 99%

On the Edgeworth expansion and the M out of N bootstrap accuracy for a Studentized trimmed mean

Gribkova

Helmers²

2007

Math. Meth. Stat.

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“…Mason and Shorack [23] obtained the necessary and sufficient conditions for the asymptotic normality of the trimmed L-statistics. The Berry -Esseen type bounds under different sets of conditions were obtained by Bjerve [4], Helmers [20]- [21], Gribkova [11]. A great contribution to the research of second order asymptotic properties for L-statistic was done by Helmers [18]- [21], who established the Edgeworth expansions for the (trimmed) L-statistics.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Cramér type large deviations for trimmed L-statistics

Gribkova

2018

pms

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In this paper, we propose a new approach to the investigation of asymptotic properties of trimmed L-statistics and we apply it to the Cramér type large deviation problem. Our results can be compared with ones in Callaert et al. (1982) the first and, as far as we know, the single article, where some results on probabilities of large deviations for the trimmed L-statistics were obtained, but under some strict and unnatural conditions. Our approach is to approximate the trimmed L-statistic by a non-trimmed L-statistic (with smooth weight function) based on Winsorized random variables. Using this method, we establish the Cramér type large deviation results for the trimmed L-statistics under quite mild and natural conditions.

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“…So the smoothness assumption in corollary 1.2 is slightly excessive for obtaining of the Berry -Esseen type bound (1.22) (cf. for instance, [14], where the optimal bound was obtained under a somewhat weaker smoothness assumption that F −1 satisfies a Lipschitz condition on the sets U a and U b , by an application of Theorem 1.1 of van Zwet [33] for symmetric statistics).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Second Order Approximations for Slightly Trimmed Means

Gribkova¹,

Helmers

2014

Theory Probab. Appl.

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We investigate the second order asymptotic behavior of distributions of statistics Tn = 1 n n−mn i=kn+1 X i:n , where kn, mn are sequences of integers, 0 kn < n − mn n, and rn := min(kn, mn) → ∞ as n → ∞, and the X i:n 's denote the order statistics corresponding to a sample X 1 , . . . , Xn of n independent identically distributed random variables. In particular, we focus on the case of slightly trimmed means with vanishing trimming percentages; i.e., we assume that max(kn, mn)/n → 0 as n → ∞, and heavy tailed distribution F ; i.e., the common distribution of the observations F is assumed to have an infinite variance. We derive optimal bounds of Berry-Esseen type of the order O(r −1/2 n ) for the normal approximation to Tn 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 Csörgő, Haeusler, and Mason [Ann. Probab., 16 (1988), pp. 672-699] and on second order approximations for (Studentized) trimmed means with fixed trimming percentages by Gribkova and Helmers [Math.

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On Analogues of Berry–Esseen Inequality for Truncated Linear Combinations of Order Statistics

Cited by 7 publications

References 4 publications

On the Edgeworth expansion and the M out of N bootstrap accuracy for a Studentized trimmed mean

On the Edgeworth expansion and the M out of N bootstrap accuracy for a Studentized trimmed mean

Cramér type large deviations for trimmed L-statistics

Second Order Approximations for Slightly Trimmed Means

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