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密度估计中的分布无关下界AI 解读

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…

自动核密度估计的一致性AI 解读

We consider the Parzen-Rosenblatt kernel density estimate on $\mathbb{R}^d$ with data-dependent smoothing factor. Sufficient conditions on the asymptotic behavior of the smoothing factor are given under which the estimate is pointwise consistent almost everywhere for all densities $f$ to be estimated. When the smoothing factor is a function only of the sample size $n$, it is shown that these conditions are also necessary, a generalization of results by Deheuvels. The…