Density estimation in RKHS with application to Korobov spaces in high dimensions

Kazashi, Yoshihito and Nobile, Fabio (2023) Density estimation in RKHS with application to Korobov spaces in high dimensions. SIAM Journal on Numerical Analysis, 61 (2). pp. 1080-1102. ISSN 0036-1429 (https://doi.org/10.1137/22M147476X)

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Abstract

A kernel method for estimating a probability density function from an independent and identically distributed sample drawn from such density is presented. Our estimator is a linear combination of kernel functions, the coefficients of which are determined by a linear equation. An error analysis for the mean integrated squared error is established in a general reproducing kernel Hilbert space setting. The theory developed is then applied to estimate probability density functions belonging to weighted Korobov spaces, for which a dimension-independent convergence rate is established. Under a suitable smoothness assumption, our method attains a rate arbitrarily close to the optimal rate. Numerical results support our theory.

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