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A multivariate normal approximation for the Dirichlet density and some applications

Ouimet, Frédéric (2021) A multivariate normal approximation for the Dirichlet density and some applications. Stat . ISSN 2049-1573. doi:10.1002/sta4.410. (In Press) https://resolver.caltech.edu/CaltechAUTHORS:20210901-153909471

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Abstract

In this short note, we prove an asymptotic expansion for the ratio of the Dirichlet density to the multivariate normal density with the same mean and covariance matrix. The expansion is then used to derive an upper bound on the total variation between the corresponding probability measures and rederive the asymptotic variance of the Dirichlet kernel estimators introduced by Aitchison and Lauder (1985) and studied theoretically in Ouimet (2020). Another potential application related to the asymptotic equivalence between the Gaussian variance regression problem and the Gaussian white noise problem is briefly mentioned but left open for future research.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1002/sta4.410DOIArticle
https://arxiv.org/abs/2103.02853arXivDiscussion Paper
ORCID:
AuthorORCID
Ouimet, Frédéric0000-0001-7933-5265
Alternate Title:Upper bound on the total variation between Dirichlet and multivariate normal distributions
Additional Information:© 2021 Wiley. Accepted manuscript online: 24 August 2021; Manuscript accepted: 10 August 2021; Manuscript revised: 28 July 2021; Manuscript received: 22 June 2021. The author acknowledges support of a postdoctoral fellowship from the NSERC (PDF) and the FRQNT (B3X supplement). We thank the referees for their valuable comments that led to improvements in the presentation of this paper. Data Availability Statement: The R code that generated all the figures in Appendix B is available as supplemental material online.
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:Dirichlet distribution; asymptotic statistics; expansion; normal approximation; Gaussian approximation; multivariate normal; total variation; asymptotic variance; nonparametric statistics; smoothing; density estimation
DOI:10.1002/sta4.410
Record Number:CaltechAUTHORS:20210901-153909471
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20210901-153909471
Official Citation:Ouimet, F. A multivariate normal approximation for the Dirichlet density and some applications. Stat. 2021;e410. Accepted Author Manuscript. https://doi.org/10.1002/sta4.410
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:110688
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:01 Sep 2021 17:42
Last Modified:01 Sep 2021 17:42

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