条件密度随机窗宽双重核估计的相合

在统计估计及检验中,有时涉及到基于样本(X_1,Y_1),…,(X_n,Y_n)直接估计未知条件密度f(yx)的不少参数方法,而双重核估计就是较常用的一种.许多学者已讨论了f(yx)的双重核估计量f_n(yx)的相合性及收敛速度,但从应用角度来看,此估计量仍有不便之处.在密度估计的情形,Devroye提出了“自动选择窗宽”核估计的概念,即窗宽完全由样本确定.本文推广这种思想于上述估计量,讨论其一致强相合性,并作为应用,顺便得到其“近邻估计”的强相合性.

THE CONSISTENCY OF THE DOUBLE KERNEL ESTIMATE OF CONDITIONAL DENSITY WITH RANDOM WINDOW WIDTH

In statistic esimation and testing, we somtimes estimate directly the unknown conditional density f(y'x) by the examples (X1, Yl), (Xn. Yn), and there are a lot of nonparametric techniques to do so, out of which the double kernel is a commonly used one. From the point of application, the double kernel estimator has many defects, for example,the choice of its window width (an)), (bn) has to be redesign. Devroye l] post the ideal of "automatively choicing window width" kernel estimator in the case of density estimation, that is to say, the window width is determined totaly by the samples themselve. we extend this idea to estimator in this article, discuss the strong consistency of it under the meaning of uniform distance, and,obtain the "nearest nighbour estimator"s consistency by the way.