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Asymptotic properties of Dirichlet kernel density estimators

Ouimet, Frédéric and Tolosana-Delgadoc, Raimon (2022) Asymptotic properties of Dirichlet kernel density estimators. Journal of Multivariate Analysis, 187 . Art. No. 104832. ISSN 0047-259X. doi:10.1016/j.jmva.2021.104832.

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We study theoretically, for the first time, the Dirichlet kernel estimator introduced by Aitchison and Lauder (1985) for the estimation of multivariate densities supported on the d-dimensional simplex. The simplex is an important case as it is the natural domain of compositional data and has been neglected in the literature on asymmetric kernels. The Dirichlet kernel estimator, which generalizes the (non-modified) unidimensional Beta kernel estimator from Chen (1999), is free of boundary bias and non-negative everywhere on the simplex. We show that it achieves the optimal convergence rate O(n^(−4/(d+4))) for the mean squared error and the mean integrated squared error, we prove its asymptotic normality and uniform strong consistency, and we also find an asymptotic expression for the mean integrated absolute error. To illustrate the Dirichlet kernel method and its favorable boundary properties, we present a case study on minerals processing.

Item Type:Article
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URLURL TypeDescription Paper
Ouimet, Frédéric0000-0001-7933-5265
Tolosana-Delgadoc, Raimon0000-0001-9847-0462
Alternate Title:Density estimation using Dirichlet kernels
Additional Information:© 2021 Elsevier Inc. Received 17 March 2021, Revised 14 August 2021, Accepted 5 September 2021, Available online 17 September 2021. F. Ouimet is supported by a postdoctoral fellowship from the NSERC (PDF) and the FRQNT (B3X supplement). We thank the Editor, the Associate Editor and the referees for their insightful remarks which led to improvements in the presentation of this paper. CRediT authorship contribution statement: Frédéric Ouimet: Writing - original draft, Writing - review & editing, Review of the literature, Conceptualization, Theoretical results and proofs, Responsible for Sections 2, 3, 4 and 6, and parts of Section 1. Raimon Tolosana-Delgado: Writing of the case study and the practical motivations in the introduction, Responsible for Section 5 and parts of Section 1.
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 kernel; Beta kernel; Asymmetric kernel; Density estimation; Simplex; Boundary bias; Variance; Mean squared error; Mean integrated absolute error; Asymptotic normality; Strong consistency; Multivariate associated kernel
Classification Code:AMS 2010 subject classifications: primary 62G07; secondary 62G05, 62G20
Record Number:CaltechAUTHORS:20200309-104903956
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Official Citation:Frédéric Ouimet, Raimon Tolosana-Delgado, Asymptotic properties of Dirichlet kernel density estimators, Journal of Multivariate Analysis, Volume 187, 2022, 104832, ISSN 0047-259X,
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:101772
Deposited By: Joy Painter
Deposited On:09 Mar 2020 17:56
Last Modified:12 Oct 2021 20:24

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