序贯最近邻密度估计

SEQUENTIAL NEAREST NEIGHBOUR DENSITY ESTIMATION

§1.引言设 X1,…,Xn是从某个具有分布 F 和密度 f 的一维总体中抽出的独立同分布（iid）样本,为估计密度函数 f（x）,1965年 Loftsgarden 和 Quesenberry 提出了下面一种估计方法:选定一个与 n 有关的自然数 k?≤n,找出最小的αn（x）,使得区间[x-（?）),x+αn（x）]内至少包含样本 X1,…,Xn中的 kn个点。

In this paper the sequential nearest neighbour density estimate f_n=(k_n-1)/(2na_n(x)) ofthe density function f(x) is considered.Stopping variable N(d) based on the ideasof fixed-width interval estimation is defined.Under proper conditions N(d) is shownto be closed and is shown to have fini finite moments of all orders.N(d) has anasymptotic normal distribution as d nconverges to zero.For such an N(d) we give anasymptotic confidence interval for f(x) of fixed-width (2d) and prescribed coverageprobability (1-α).