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- W2035644590 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 $f:inf_{h > 0}n^{2/5}J_n geq CA(K)B^ast(f) + o(1) geq 0.6076703cdots + o(1)$ where $C = 1.028493cdots$ 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 $C^ast = 1.3768102cdots$ is another universal constant. For this class of densities, we also obtain the exact asymptotic behavior of $J_n$." @default.
- W2035644590 created "2016-06-24" @default.
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- W2035644590 date "1984-12-01" @default.
- W2035644590 modified "2023-10-17" @default.
- W2035644590 title "Distribution-Free Lower Bounds in Density Estimation" @default.
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- W2035644590 doi "https://doi.org/10.1214/aos/1176346790" @default.
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