Asymptotic properties of Dirichlet kernel density estimators

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.

25 pages, 3 figures; v4: final version

References in corpus (9)

• Asymptotic properties of Bernstein estimators on the simplex
• A precise local limit theorem for the multinomial distribution and some applications
• On the Le Cam distance between Poisson and Gaussian experiments and the asymptotic properties of Szasz estimators
• A study of seven asymmetric kernels for the estimation of cumulative distribution functions
• A symmetric matrix-variate normal local approximation for the Wishart distribution and some applications
• Smooth Distribution Function Estimation for Lifetime Distributions using Szasz-Mirakyan Operators
• On the boundary properties of Bernstein estimators on the simplex
• A multivariate normal approximation for the Dirichlet density and some applications
• Gamma kernel estimation of the density derivative on the positive semi-axis by dependent data

Cited by in corpus (11)

• Asymptotic properties of Bernstein estimators on the simplex
• General formulas for the central and non-central moments of the multinomial distribution
• A study of seven asymmetric kernels for the estimation of cumulative distribution functions
• A symmetric matrix-variate normal local approximation for the Wishart distribution and some applications
• On the boundary properties of Bernstein estimators on the simplex
• Minimax properties of Dirichlet kernel density estimators
• Local linear smoothing for regression surfaces on the simplex using Dirichlet kernels
• Multiple combined gamma kernel estimations for nonnegative data with Bayesian adaptive bandwidths
• Explicit formulas for the joint third and fourth central moments of the multinomial distribution
• Bayesian Bandwidths in Semiparametric Modelling for Nonnegative Orthant Data with Diagnostics
• Normal approximations for the multivariate inverse Gaussian distribution and asymmetric kernel smoothing on $d$-dimensional half-spaces