Asymptotic properties of Dirichlet kernel density estimators

Abstract

In this paper, we introduce Dirichlet kernels for the estimation of multivariate densities supported on the d-dimensional simplex. These kernels generalize the beta kernels from Brown & Chen (1999); Chen (1999, 2000a); Bouezmarni & Rolin (2003), originally studied in the context of smoothing for regression curves. We prove various asymptotic properties for the estimator : bias, variance, mean squared error, mean integrated squared error, asymptotic normality and uniform strong consistency. In particular, the asymptotic normality and uniform strong consistency results are completely new, even for the case d = 1 (beta kernels). These new kernel smoothers can be used for density estimation of compositional data. The estimator is simple to use, free of boundary bias, allocates non-negative weights everywhere on the simplex, and achieves the optimal convergence rate of n −4/(d+4) for the mean integrated squared error.

A symmetric matrix-variate normal local approximation for the Wishart distribution and some applications

“…The asymptotics of the pointwise bias and variance were first computed by Chen [15,16] for Beta and Gamma kernel estimators, by Ouimet and Tolosana-Delgado [43] for the Dirichlet kernel estimator of Aitchison and Lauder [2], and by Kokonendji and Somé [35,36] for multivariate associated kernel estimators. The next two theorems below extend the (unmodified) Gamma case to our multidimensional setting.…”

Multiple combined gamma kernel estimations for nonnegative data with Bayesian adaptive bandwidths

“…Asymmetric kernels are known to improve smoothing quality for partially or totally bounded supports; e.g. Scaillet [33] with inverse and reciprocal inverse Gaussian kernels, [30] using Dirichlet kernels and, [27] and [43] for uni-and multivariate generalized Birnbaum-Saunders, see also Kakizawa [19]. However, these kernels induce an additional quantity in the bias that needs reduction via modified versions.…”

A Study of Seven Asymmetric Kernels for the Estimation of Cumulative Distribution Functions

“…Hirukawa [15], Bouezmarni and Rombouts [16], Zhang and Karunamuni [17], Bertin and Klutchnikoff [18,19], Igarashi [20]; • Gamma, inverse Gamma, LogNormal, inverse Gaussian, reciprocal inverse Gaussian, Birnbaum-Saunders and Weibull kernels, when the target density is supported on [0, ∞), see, e.g., Chen [3], Jin and Kawczak [21], Scaillet [22], Bouezmarni and Scaillet [23], Fernandes and Monteiro [14], Bouezmarni and Rombouts [16,24,25], Igarashi and Kakizawa [26,27], Charpentier and Flachaire [28], Igarashi [29], Zougab and Adjabi [30], Kakizawa and Igarashi [31], Kakizawa [32], Zougab et al [33], Zhang [34], Kakizawa [35]; • Dirichlet kernel, when the target density is supported on the d-dimensional unit simplex, see [1] and the first theoretical study by Ouimet and Tolosana-Delgado [36]. • Continuous associated kernels, the aim of which is to unify the theory of asymmetric kernels with the one for traditional kernels in both the univariate and multivariate settings, see, e.g., Kokonendji and Libengué Dobélé-Kpoka [37], Kokonendji and Somé [38,39].…”

“…The interested reader is referred to Hirukawa [40] and Section 2 of Ouimet and Tolosana-Delgado [36] for a review of some of these papers and an extensive list of papers dealing with asymmetric kernels in other settings.…”