On arbitrarily slow rates of global convergence in density estimation
On arbitrarily slow rates of global convergence in density estimation
复制标题
关于密度估计中任意缓慢的全局收敛速率
DOI：
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发表时间：
1983
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通讯作者：
L. Devroye
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文献类型：
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作者：
L. Devroye
设Rd上的密度f用fn（x, X1，…）， Xn)其中x∈Rd， fn是其参数的Borel可测函数，X1，…， Xn是具有公共密度f的独立随机向量，设p≧1为常数。本笔记的一个主要结果是，对于每一个数列fn，以及每一个正数数列且满足lim =0，存在一个f，使得
$$Eleft( {smallint |f_n left( x
ight) - fleft( x
ight)|^p dx}
ight) > a_n$$
无限频繁。在这里，只要看一下所有以2为界并消失在[0,1]d之外的f就足够了。对于p=1， f总是可以被限制为无穷多次连续可微密度，所有导数都是绝对有界和绝对可积的。
SummaryLet a density f on Rd be estimated by fn(x, X1, ..., Xn) where x∈Rd, fn is a Borel measurable function of its arguments, and X1, ..., Xn are independent random vectors with common density f. Let p≧1 be a constant. One of the main results of this note is that for every sequence fn, and for every positive number sequence an satisfying lim an=0, there exists an f such that
$$Eleft( {smallint |f_n left( x
ight) - fleft( x
ight)|^p dx}
ight) > a_n$$
infinitely often.Here it suffices to look at all the f that are bounded by 2 and vanish outside [0, 1]d. For p=1, f can always be restricted to the class of infinitely many times continuously differentiable densities with all derivatives absolutely bounded and absolutely integrable.