Convergence Rates Of Nearest Neighbor Density Estimators Under NA Sample

The concept of negatively associated(NA) sequence was proposed firstly by Joag-Dey and Proschan(1983) and Block and Savits(1982). Joag-Dey and Proschan discussed some basic prop-erties and practical applications of NA sequences. Roussas(1994) and Beak(2003) also studied the basic properties of NA sequences such as complete convergence. Due to the broad applications of NA sequences, the statistical inferences associated with NA sequences have attracted great atten-tions of many scholars. Large sample properties of NA sequences such as the asymptotic normality of kernel density estimation have been investigated extensively.Nearest neighbor density estimator is an important non-parametric density estimation method, which was proposed by Loftsgarden and Quesenberry in 1965. They proved the weak consistency of nearest neighbor density estimator f_n(x). Since the concept of nearest neighbor density esti-mator was introduced, many scholars studied its characteristics such as consistency and uniform convergence under all kinds of samples and various conditions. As a result, a lot of good results have been obtained in this aspect.According to the idea of nearest neighbor estimator, Yu proposed a new nearest neighbor density estimator f_n(x) based on order statistics in 1986. Under independent samples and mild conditions, he established its pointwise weak and strong consistency, uniform weak and strong consistency and the L1-mold strong consistency on a bounded interval. Xue (1992,1994) con-sidered the consistency and convergence rates of f_n(x) under independent and φ-mixing samples respectively. This paper mainly studies the pointwise consistency and strong uniform consistency, and obtain the convergence rates of pointwise consistency and uniform strong consistency of f_n(x) respectively.Here we summary the mainly new findings and innovations in this paper:1. This paper firstly proves the consistency of nearest neighbor density estimator f_n(x) and obtains the convergence rates under NA sample.2. The method from this paper can provide a good reference for studying large sample char-acteristics of f_n(x) under other dependent samples.