When does Edgeworth beat Berry and Esséen? Numerical evaluations of Edgeworth expansions
✍ Scribed by Munsup Seoh; Marc Hallin
- Book ID
- 104340357
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 889 KB
- Volume
- 63
- Category
- Article
- ISSN
- 0378-3758
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✦ Synopsis
It has been common practice to derive higher-order expansions with remainder terms R,, expressed under O(-) or o(.) forms, then presenting simulations of small sample sizes for comparisons with the asymptotic results. Such comparisons are intended to support the idea that the unknown constants hidden behind the O(.) or o(.) remainders are reasonably small. That practice however is marred by an obvious logical flaw. While a simulation study for fixed sample size no can be used in assessing the magnitude of the remainder term R,, for n=n~, it contains no information about the behavior of R, fbr n 74 no and is definitely irrelevant in the evaluation of such quantities as sup,,(n~R,,). In particular, the common belief that the remainder term R,, is monotonically decreasing with n seems to be totally wrong. The particular case of Edgeworth expansions (to the o(n -r) order) is considered here. Such expansions are expected to improve on traditional O(n-L:2) Berry-Ess6en bounds. Nobody however seems to have considered the most natural question we are addressing here: how large should n be .fi~r Edqeworth beating, t Berry and EssOen? In the special case of the Wilcoxon signed rank statistic, a strict numerical translation (Hallin and Seoh, 1996) of the only available convergence result (A[bers et al. Ann. Statist. 4 (1976) 108-156) leads to the rather distressing conclusion that Edgeworth expansions do not improve upon Berry-Ess6en bounds until the sample size reaches the astronomically huge value of e s°°° ! Now, the objective of Albers ct al. was not to obtain good numerical values for their remainder terms. We therefore revisit their results and, instead of considering monotonically decreasing bounds (on the remainder term) of the form An-~,~rj "_ where A is an appropriate, universal constant (on the model of Berry-Ess6en's), we rather derive non-monotonic bounds of the form A,,n -~'=j~'2, with an explicit, closed-form (and unifbrmly hounded) sequence of constants A, (n denotes the sample size). Under this approach, the sampie size at which Edgeworth beats Berry and Ess6cn drops considerably, from e 5°°° to 128 098. Unfortunately, 128098 hardly can be considered as a small nor as a moderately large sample * Coressponding author.