Density estimation using Dirichlet kernels
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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.
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