多维近邻密度估计的L_P模强相合性

Strong L_p-consistency of Multidimensional Nearest Neighbor Density Estimates

设X_1,…,X_n是i.i.d.的具有密度f(x)的d维随机变量。设S_(x,a(x))是中心在x且至少包含X_1,…,X_n中k_n个点的最小的球,则f_n(x)=R_n/(n|S_(x,a(x)|)是f(x)一个近邻估计。我们证明了:假如lim k_n/n=0,lim k_n/logn=∞以及flog^+f在任何有界Borel集上可积(或∫f'(x)dx<∞,p>1),则对任何有界Borel集A有(或p>1)。反之,如,则有∫f^p(x)dx<∞,lim n→∞k_n/n=0和lim n→∞k_n=∞。

Suppose that X_1,…,X_n are i. i. d. R^d-valued random variables with adensity f(x). If S_(x.a(x)) is the smallest sphere centered at x and containing atleast k. points of X_1,…, X_n, then f_n(x) = k_n/n|S_(n,a(x))| is nearest neighbor density estimate of f(x). We proved that if lim k_n/n = 0, lim k_n/ log n =∞ andflog+f is L-integrable on any bounded Borel set (or∫f^p(x)dx<∞, p>1), thenlim∫_A|f_n(x) -f(x)|dx?0 for any bounded Borel set A(or lim∫|f_n(x)-f(x)| dx?0, p>1). Conversely, if lim∫|f_n(x)- f(x) | ~pdx?0, then∫f^p(x)dx<∞, lim k_n/n= 0 and lim k_n= ∞.