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Asymptotic properties of Bernstein estimators on the simplex

Ouimet, Frédéric (2021) Asymptotic properties of Bernstein estimators on the simplex. Journal of Multivariate Analysis, 185 . Art. No. 104784. ISSN 0047-259X. doi:10.1016/j.jmva.2021.104784. https://resolver.caltech.edu/CaltechAUTHORS:20200309-105256028

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Abstract

Bernstein estimators are well-known to avoid the boundary bias problem of traditional kernel estimators. The theoretical properties of these estimators have been studied extensively on compact intervals and hypercubes, but never on the simplex, except for the mean squared error of the density estimator in Tenbusch (1994) when d=2. The simplex is an important case as it is the natural domain of compositional data. In this paper, we make an effort to prove several asymptotic results (bias, variance, mean squared error (MSE), mean integrated squared error (MISE), asymptotic normality, uniform strong consistency) for Bernstein estimators of cumulative distribution functions and density functions on the d-dimensional simplex. Our results generalize the ones in Leblanc (2012a) and Babu et al. (2002), who treated the case d=1, and significantly extend those found in Tenbusch (1994). In particular, our rates of convergence for the MSE and MISE are optimal.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1016/j.jmva.2021.104784DOIArticle
https://arxiv.org/abs/2002.07758arXivDiscussion Paper
ORCID:
AuthorORCID
Ouimet, Frédéric0000-0001-7933-5265
Additional Information:© 2021 Elsevier Inc. Received 24 February 2021, Revised 23 June 2021, Accepted 24 June 2021, Available online 5 July 2021. The author is supported by a postdoctoral fellowship from the NSERC (PDF) and the FRQNT (B3X supplement). We thank the Editor, Associate Editor and referees, as well as our financial sponsors. CRediT authorship contribution statement: Frédéric Ouimet: Conceptualization, Writing, Proofs.
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:Asymptotic normality; Bernstein estimators; Compositional data; Cumulative distribution function estimation; Density estimation; Mean squared error; Simplex; Uniform strong consistency
Classification Code:AMS 2020 Subject Classifications: Primary: 62G05; Secondary: 62G07, 62G20, 60F05
DOI:10.1016/j.jmva.2021.104784
Record Number:CaltechAUTHORS:20200309-105256028
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20200309-105256028
Official Citation:Frédéric Ouimet, Asymptotic properties of Bernstein estimators on the simplex, Journal of Multivariate Analysis, Volume 185, 2021, 104784, ISSN 0047-259X, https://doi.org/10.1016/j.jmva.2021.104784.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:101773
Collection:CaltechAUTHORS
Deposited By: Joy Painter
Deposited On:09 Mar 2020 20:44
Last Modified:19 Aug 2021 21:30

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