Title: Posterior Contraction Rates of Density Derivative Estimation

Author(s): Weining Shen and Subhashis Ghosal
Issue: Volume 79 Series A Part 2 Year 2017
Pages: 336 -- 354
In this paper, we study the problem of Bayesian estimation of derivatives of a density function on the unit interval. We use a finite random series prior based on $B$-splines and study the asymptotic properties of the posterior distribution under the setting of fixed smoothness of the true function. We obtain the posterior contraction rate under both the $L_2$- and $L_{\infty}$-distances. The rate under $L_2$-distance agrees with the minimax optimal rate. This result is then extended to the estimation of a multivariate density function on the unit cube and its mixed partial derivatives using tensor product $B$-splines.
AMS (2000) subject classification . Primary: 62G20, Secondary: 62G05.
Keywords and phrases: B-spline, Density derivative estimation, Nonparametric Bayes, Posterior contraction rate, Tensor product.