Convergence Properties For END Random Variables And Its Application In The Estimation Of Density Function

With the progress of society,the probability limit theory is used more and more.The combination with economy,health care and education is getting closer and closer.However,most of the sample data obtained in real life are not independent,so more and more scholars study the properties of dependent variables.This paper mainly studies the complete moment convergence and complete integral convergence of the extended negative dependent variables(for short,END)and do a numerical simulation.Then we extend the END random variables to the widely orthant dependent(for short,WOD)random variables and study the asymptotic properties for the estimators of the survival function and failure function based on WOD samples.The first chapter mainly introduces the background of this paper and the concept of dependent random variables.In the second chapter,we mainly introduce the inequalities and lemmas needed in the process of proving the relevant conclusions of this paper.In the third chapter,we generalize some results of negatively associated random variables(for short,NA)and negative orthant dependent random variables(for short,NOD),obtain the complete moment convergence and integral convergence for the maximum sequence of END random variables,and give some equivalent propositions of complete moment convergence and complete integral convergence for the sequence of END random variables.In the fourth chapter,the consistency for the estimators of the survival function and failure rate function in reliability theory is investigated.The strong consistency and the convergence rate for the estimators of the survival function and failure rate function based on WOD samples are established.Our results established in the paper generalize the corresponding ones for independent samples and some negatively dependent samples.In the fifth chapter,we will carry out a numerical simulation to study the Marcinkiewicz-Zygmund type strong laws of large numbers for END random variables.In the sixth chapter,we summarize some shortcomings and innovations of this paper.