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

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.