Complete Consistency Of Density Estimation For M-WOD Random Variables

As a branch of statistics,probability limit theory is very important,and one of the main problems in this aspect is the estimation of probability density functions.At first,people conducted a lot of research on the estimation of probability density functions for independent sequences.But in reality,the condition of independence is difficult to fit.So many scholars began to pay attention to dependent sequences.There are many results on dependent sequences up to now.Kernel density estimator and nearest neighbor estimator of density function are two popular density estimators,which have been used in many scenarios since their emergence.Recursive kernel density estimator is a density estimator based on kernel density estimator,which is faster than kernel density estimator in large sample sizes.Therefore,many scholars have paid attention to it.Based on previous research,this paper studies the complete consistency of recursive kernel density estimator and nearest neighbor density estimator for m-WOD sequences.The main content of this paper is divided into the following two parts:The first part studies the complete consistency of recursive kernel density estimator for m-WOD random variable sequences.Firstly,by limiting kernel functions and bandwidths,a key inequality is proved using Taylor expansion of the function and Lebesgue’s Dominated Convergence Theorem.On this basis,the complete consistency of the recursive kernel density estimator for m-WOD random variable sequences under different conditions is obtained.Then,according to the definition of the hazard rate function,the complete consistency of the estimator of the hazard rate function under different conditions is obtained by using the Bernstein-type inequality of m-WOD random variable sequences.The second part studies the complete consistency of nearest neighbor density estimator for m-WOD random variables.Using the properties of locally Lipschitz functions and Lagrange Mean Value Theorem,a key inequality is obtained.Then using the Bernstein-type inequality,the complete consistency of nearest neighbor density estimator for the m-WOD random variables under certain conditions is obtained.The limit theory of nearest neighbor density estimator is generalized.The m-WOD random variable sequences are a wide range of sequences with many different applications in the fields of economics,computers,astronomy,etc.Recursive kernel density estimator and nearest neighbor density estimator have the advantages of good results and simple principles,which are used in many practical problems,so it is necessary to study the properties of these two types of density estimation for m-WOD sequences.