Plug-in error bounds for a mixing density estimate in $R^d,$ and for its derivatives

TL;DR

This paper derives plug-in error bounds for estimating a mixing density and its derivatives in multivariate settings, relating the bounds to the density estimate's error, kernel properties, and bandwidth.

Contribution

It provides new upper bounds for the error of mixing density estimates in multiple dimensions, extending previous one-dimensional results to higher dimensions.

Findings

Bounds depend on the kernel's Fourier transform and bandwidth.

Error rates are nearly optimal when the density estimate is optimal.

Bounds vary with the smoothness of the kernel and the density.

Abstract

A mixture density, $f_p,$ is estimable in $R^d, \ d \ge 1,$ but an estimate for the mixing density, $p,$ is usually obtained only when $d$ is unity; $h$ is the mixture's kernel. When $f_p$'s estimate has form $f_{\hat p_n}$ and $p$ is $\tilde q$-smooth, vanishing outside a compact in $R^d,$ plug-in upper bounds are obtained herein for the $L_u$-error (and risk)of $\hat p_n$ and its derivatives; $d \ge 1, 1 \le u \le \infty.$ The bounds depend on $f_{\hat p_n}$'s $L_u$-error (or risk), $h$'s Fourier transform, $\tilde h,$ and the bandwidth of kernel $K$ used in approximations. The choice of $\hat p_n,$ via $f_{\hat p_n},$ suggests that $\hat p_n$'s error rate could be only nearly optimal when $f_{\hat p_n}$ is optimal, but competing estimates and their error rates may not be available for $d>1.$ In examples with $d$ unity, the upper bound is optimal when $h$ is super smooth, misses the optimal rate by the factor $(\log n)^{\xi}, \ \xi>0,$ when $h$ is smooth, and is satisfactory when $\tilde h$ has periodic zeros.