# Density estimation using Dirichlet kernels

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), Chen (2000), 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.

02/18/2020

### Asymptotic properties of Bernstein estimators on the simplex

In this paper, we study various asymptotic properties (bias, variance, m...
01/25/2018

### Bivariate density estimation using normal-gamma kernel with application to astronomy

We consider the problem of estimation of a bivariate density function wi...
05/31/2019

### Targeted Estimation of L2 Distance Between Densities and its Application to Geo-spatial Data

We examine the integrated squared difference, also known as the L2 dista...
05/04/2018

### Axiomatic Approach to Variable Kernel Density Estimation

Variable kernel density estimation allows the approximation of a probabi...
12/07/2011

### A recursive procedure for density estimation on the binary hypercube

This paper describes a recursive estimation procedure for multivariate b...
07/13/2020

### Strong Uniform Consistency with Rates for Kernel Density Estimators with General Kernels on Manifolds

We provide a strong uniform consistency result with the convergence rate...
11/30/2020

### A study of seven asymmetric kernels for the estimation of cumulative distribution functions

In Mombeni et al. (2019), Birnbaum-Saunders and Weibull kernel estimator...