Edgeworth Expansions for a Sample Sum from a Finite Set of Independent Random Variables

Hu, Zhishui; Robinson, John; Wang, Qiying

doi:10.1214/ejp.v12-447

Cited by 4 publications

(13 citation statements)

References 17 publications

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“…We obtain here a three-term asymptotic expansion under very general conditions on the parameters, generalizing the results in Mirakhmedov (1992) and Ivchenko and Mirakhmedov (1992). The result in (ii) improves the main results of Mirakhmedov (1983), Bloznelis (2000), as well as parts of Theorem 1 of Hu et al (2007).…”

Section: Introductionsupporting

confidence: 82%

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On Edgeworth Expansions in Generalized Urn Models

2012

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Abstract. The random vector of frequencies in a generalized urn model can be viewed as conditionally independent random variables, given their sum. Such a representation is exploited here to derive Edgeworth expansions for a "sum of functions of such frequencies", which are also called "decomposable statistics." Applying these results to urn models such as with-and withoutreplacement sampling schemes as well as the multicolor Pólya-Egenberger model, new results are obtained for the chi-square statistic, for the sample sum in a without replacement scheme, and for the so-called Dixon statistic that is useful in comparing 2 samples.

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Section: Introductionsupporting

confidence: 82%

“…Also, a very special case of (2.5) is the most commonly used formula of Erdös and Renyi (1959) for investigating the sample sum in a without-replacement scheme (see e.g. Babu and Singh (1985), Zhao et al (2004) and Hu et al (2007)). Formula (2.3) is also useful in studying large deviation problems (see e.g.…”

Section: Introductionmentioning

confidence: 99%

On Edgeworth Expansions in Generalized Urn Models

2012

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“…We wish to derive a suitable upper bound for

sup_{u \in ℝ} |P ({}, \frac{f_{1} false({\hat{Y}}_{normalts}^{'} - Y false)}{n^{1 / 2} σ_{normalts}} < u) - Ψ_{ts} (u)|,

by utilizing Theorem 2 of Hu, Robinson, & Wang (). For doing this, we will need a technical condition, which ensures that the values

y_{jk}

in each secondary unit do not cluster around too few values (cf.…”

Section: Estimators In Two‐stage Samplingmentioning

confidence: 99%

“…As we show in Section the estimator under a two‐stage scheme can be reduced to a weighted sample mean from a finite population of “random” variables. Asymptotic normality and Edgeworth asymptotic expansion for this sample mean have been considered by von Bahr (), Mirakhmedov (), Hu, Robinson, & Wang, (), Mirakhmedov, Jammalamadaka, & Ibrahim (), and Ibrahim & Mirakhmedov ().…”

Section: Introductionmentioning

confidence: 99%

Edgeworth expansions for two‐stage sampling with applications to stratified and cluster sampling

2015

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A two-term Edgeworth expansion for the standardized version of the sample total in a two-stage sampling design is derived. In particular, for the commonly used stratified and cluster sampling schemes, formal two-term asymptotic expansions are obtained for the Studentized versions of the sample total. These results are applied in conjunction with the bootstrap to construct more accurate confidence intervals for the unknown population total in such sampling schemes. Résumé: Les auteurs présentent un développement en deux termes d'une série d'Edgeworth pour l'estimateur du total basé sur un planéchantillonnalà deux phases. Ils obtiennent en particulier des développements asymptotiques formels pour la version studentisée du totaléchantillonnal basé sur unéchantillonnage stratifié et sur unéchantillonnage en grappes. Les auteurs utilisent ces résultats et des méthodes de rééchantillonnage afin de construire des intervalles de confiance plus précis pour le total de la population dans le contexte de ces plans d'échantillonnage. La revue canadienne de statistique 43: 578-599; 2015

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“…Von Bahr's bound has been improved in Corollary of Mirakhmedov (1985) and in Theorem 1 of Zhao et al (2004). Two terms Edgeworth asymptotic expansion result has been established by Mirakhmedov (1979) and Hu et al (2007a), whereas expansion with arbitrary number of terms was obtained by Mirakhmedov (1983). Large deviation results follows from Theorems 11 and 12 of Mirakhmedov (1996).…”

mentioning

confidence: 99%