最近邻估计的局部一致强收敛速度

Retes of Strong Uniform Convergence on Local Region of Nearest Neighbor Density Estimates

设 X_1,X_2,…,X_n,i.i.d.是具有概率密度函数 f(x)的随机变量,定义(x)=(na_n(x))^(-1)K((X_i-x)/α_n(x)),n≥1。本文中,我们在某些紧集 B 上讨论了了(x)一致强收敛于 f(x)的速度.若{c_n}是任意一个趋于无穷的数列,我们得到:c_n^(-1)(n/logn)^(m/(2m+1))|~n(x)-f(x)|→0,α.e.n→∞.

Let X_1,…,X.be i.i.d.r.V.with a probability density function f(x).Define _n(x)=(na_n(n))^(-1) sum from (i=1) to n K((X_i-x)/a_n(x)),n≥1. In the paper,we discuss the strong uniform convergence rate of _n(x) to f(x) on some compact set B. If {c_n} is any sequence tending to ∞,we can obtain c_n^(-1)(n/1nn)^(m/(2m+1))sup|)n(x)-f(x)|→0,a.e.,as n→∞.