Consistency Of The Density Function Estimation Under ND Or WOD Sample

It is well known that the limit theory of independent random variable sequences are one of the most important objects in probability, and has been obtained well development, in many practical problems which the samples are not independent, but have a certain connection, after that, Statisticians have proposed the concept of all kinds of dependent sequences。in1983, Joag-Dev etc introduced the concept of ND dependent sequences, and noted that the ND sequences are weaker than the NA sequences。Because of the ND sequences have a wide range of applications in reliability theory、multivariate statistical analysis、percolation theory etc, so there are many scholars to research its limit properties deeply and have made many important theoretical results。In2010, when Y.B. Wang etc researched the Uniform asymptotic for the finite-time ruin probability of a dependent risk model with a constant interest rate, they have put forward the concept of the dependency structure of the WOD sequences, which are wider than ND sequences, they not only contain negative dependency sequence, may also include some positive dependency sequence, which further enriched the theory of dependency sequence structure。Therefore, it is very meaningful to extend the large sample properties of the independent sequences or NA sequences to ND or WOD sequences。Based on the wide range of the ND and WOD sequences, this paper researched the consistency of the density function estimation under ND and WOD samples, also obtained some convergence properties about nearest neighbor density estimator and kernel density estimator under ND and WOD samples, such as the strong consistency and the mean square consistency etc, further extended the large sample properties of the independence and other dependency case。In chapter1, we introduced some background material and this paper’s research results about density function estimator under ND and WOD sequences。In chapter2, we researched the rate of strong and uniform strong consistency of the nearest neighbor density estimator under ND samples, by using exponential inequality and other properties of the ND sequences, and extended the corresponding results under NA sequences。In chapter3, the strong consistency and mean square consistency of the kernel estimator of the density function for ND samples are proved respectively under suitable conditions, by using Bernstein inequality and moment inequality。In chapter4, the strong consistency of the kernel estimator of the density function for WOD samples is proved by using WUOD exponential inequality。...