Asymptotic properties of Dirichlet kernel density estimators
Asymptotic properties of Dirichlet kernel density estimators
Ouimet, F.; Tolosana Delgado, R.
Abstract
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.
Keywords: Asymmetric kernel; Asymptotic normality; Beta kernel; Boundary bias; Density estimation; Dirichlet kernel; Mean integrated absolute error; Multivariate associated kernel; Simplex; Strong consistency
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Journal of Multivariate Analysis 187(2022), 104832
Online First (2021) DOI: 10.1016/j.jmva.2021.104832
Cited 7 times in Scopus
Permalink: https://www.hzdr.de/publications/Publ-33466