Asymptotic properties of Dirichlet kernel density estimators

Frédéric Ouimet and Raimon Tolosana-Delgado

Journal of Multivariate Analysis, 2022, vol. 187, issue C

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: 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 (search for similar items in EconPapers)
Date: 2022
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Citations: View citations in EconPapers (3)

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DOI: 10.1016/j.jmva.2021.104832

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