Convergence Rates Of Nearest Neighbor Density Estimator Under PA Samples

The concept of positive associated random variables was first proposed and researched by Esary, Proschan and Walkup in 1967. This concept has been widely used in statistical mechanic-s, penetration theory, reliability analysis and other fields. There are many scholars making deep study on PA. It is easy to see that the related properties and moment inequalities of PA random vari-ables have been discussed in the literature [1-19]. Loftsgarden and Quesenberry first introduced the nearest neighbor density estimation method in 1965. It is widely used in the Social Sciences, engineering and technical, physical sciences, etc. At present, there are many researches about the nearest neighbor density estimation. From the literature [23-43], we can see that the consis-tency and convergence rate under the independent samples and the dependent samples (including negative association, a mixed,etc.)have been studied extensively.In this paper, we study the consistency and convergence rates of the nearest neighbor density estimation under the PA samples. It is proved that the strong convergence rate of the nearest neighbor density estimator is close to n"1/4, and the uniformly strong convergence rate is almost n-1/6 for the PA samples. We further study the uniform strong consistency of the estimation of the failure rate function under the PA samples. At the same time, the nearest neighbor density estimation is made by numerical simulation, and the advantages and disadvantages of the nearest neighbor density estimation method are compared.Some new findings of this paper are summarized as follows;(1) In this paper, we first study the consistency and convergence rate of the nearest neighbor density estimator under the PA samples.(2) In this paper, we prove that the strong convergence rate and the uniform strong conver-gence rate of the nearest neighbor density estimation under the PA samples is consistent with that of the negative associatied (NA) samples.