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A precise local limit theorem for the multinomial distribution and some applications

Ouimet, Frédéric (2021) A precise local limit theorem for the multinomial distribution and some applications. Journal of Statistical Planning and Inference, 215 . pp. 218-233. ISSN 0378-3758. doi:10.1016/j.jspi.2021.03.006. https://resolver.caltech.edu/CaltechAUTHORS:20210715-152443248

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Abstract

In Siotani and Fujikoshi (1984), a precise local limit theorem for the multinomial distribution is derived by inverting the Fourier transform, where the error terms are explicit up to order N⁻¹. In this paper, we give an alternative (conceptually simpler) proof based on Stirling’s formula and a careful handling of Taylor expansions, and we show how the result can be used to approximate multinomial probabilities on most subsets of R^d. Furthermore, we discuss a recent application of the result to obtain asymptotic properties of Bernstein estimators on the simplex, we improve the main result in Carter (2002) on the Le Cam distance bound between multinomial and multivariate normal experiments while simultaneously simplifying the proof, and we mention another potential application related to finely tuned continuity corrections.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1016/j.jspi.2021.03.006DOIArticle
https://arxiv.org/abs/2001.08512arXivDiscussion Paper
ORCID:
AuthorORCID
Ouimet, Frédéric0000-0001-7933-5265
Alternate Title:A precise local limit theorem for the multinomial distribution
Additional Information:© 2021 Elsevier B.V. Received 23 January 2020, Revised 24 March 2021, Accepted 24 March 2021, Available online 1 April 2021. We thank the referees for their useful comments, in particular for bringing up the Ref. Siotani and Fujikoshi (1984). The author acknowledges support of a postdoctoral fellowship from the Natural Sciences and Engineering Research Council of Canada (PDF) and a supplement from the Fonds de Recherche du Québec - Nature et Technologies, Canada (B3X).
Funders:
Funding AgencyGrant Number
Natural Sciences and Engineering Research Council of Canada (NSERC)UNSPECIFIED
Fonds de recherche du Québec – Nature et technologies (FRQNT)UNSPECIFIED
Subject Keywords:Multinomial distribution; Local limit theorem; Multivariate normal; Bernstein estimators; Deficiency; Comparison of experiments
DOI:10.1016/j.jspi.2021.03.006
Record Number:CaltechAUTHORS:20210715-152443248
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20210715-152443248
Official Citation:Frédéric Ouimet, A precise local limit theorem for the multinomial distribution and some applications, Journal of Statistical Planning and Inference, Volume 215, 2021, Pages 218-233, ISSN 0378-3758, https://doi.org/10.1016/j.jspi.2021.03.006.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:109843
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:15 Jul 2021 16:57
Last Modified:15 Jul 2021 16:57

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