相依随机变量的密度函数的递归核估计的渐近正态性

ASYMPTOTIC NORMALITY OF RECURSIVE KERNEL DENSITY ESTIMATES UNDER DEPENDENT ASSUMPTIONS

设{X_n;n≥1}为同分布的ρ-混合序列,其未知密度,f(x)的递归核估计为: f_n(x)=1/n sum from j=1 to n h_j^(-1)K(x-X_j/h_j),本文在适当的条件下,讨论由f_n(x)所产生的随机元的有限维渐近正态性。

Let X_1, …, X_n be random samples from an unknown density funotion f(x). The recursive kernel density function estimator can be obtained by putting f_n(x)=1/n sum from j=1 to n h_j^(-1)K(x-X_j/h_j), where K is a univariate kernel funotion, and h_n is a sequence of positive numbers converging to zero. In the paper, the asymptotio multi-normality of g_n(x) is given in the case of dependent sample, where g_n(x)=(f(x)-Ef_n(x))/(var(f_n(x)))^(1/2).