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Asymptotic properties of Dirichlet kernel density estimators

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  • NSTL
  • Elsevier
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. (c) 2021 Elsevier Inc. All rights reserved.

Dirichlet kernelBeta kernelAsymmetric kernelDensity estimationSimplexBoundary biasVarianceMean squared errorMean integrated absolute errorAsymptotic normalityStrong consistencyMultivariate associated kernelMULTIPLICATIVE BIAS CORRECTIONLOCAL LINEAR-REGRESSIONCENTRAL-LIMIT-THEOREMBANDWIDTH SELECTIONASYMMETRIC KERNELSSEMIPARAMETRIC ESTIMATIONNONPARAMETRIC-ESTIMATIONBERNSTEIN POLYNOMIALSBOUNDARY CORRECTIONSMOOTH ESTIMATION

Ouimet, Frederic、Tolosana-Delgado, Raimon

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CALTECH

Helmholtz Zentrum Dresden Rossendorf

2022

10.1016/j.jmva.2021.104832

Journal of Multivariate Analysis

Journal of Multivariate Analysis

SCI
ISSN：0047-259X
年,卷(期)：2022.187
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  • 193