Distribution-Free Lower Bounds in Density Estimation

Abstract

We consider the kernel estimate on the real line, $f_n(x) = (nh)^{-1} \sum^n_{i=1} K((X_i - x)/h),$ where $K$ is a bounded even density with compact support, and $X_1, \cdots, X_n$ are independent random variables with common density $f$. We treat the problem of placing a lower bound on the $L$1 error $J_n = E(\int |f_n - f|)$ which holds for all $f$. In particular, we show that there exist $A(K) \geq (9/125)^{1/5}$ depending only upon $K$, and $B^\ast(f) \geq 1$ depending only upon $f$ such that (i) for all $f:\inf_{h > 0}n^{2/5}J_n \geq CA(K)B^\ast(f) + o(1) \geq 0.6076703\cdots + o(1)$ where $C = 1.028493\cdots$ is a universal constant; (ii) for all $f$ with compact support and two bounded continuous absolutely integrable derivatives, $\inf_{h > 0}n^{2/5}J_n \leq C^\ast A(K)B^\ast(f) + o(1)$ where $C^\ast = 1.3768102\cdots$ is another universal constant. For this class of densities, we also obtain the exact asymptotic behavior of $J_n$.