Statistical Inference Of Distribution Functions At A Number Of Finite Points Under Associated Samples

In1967, Esary et al.[1] introduced the concept of positively associated random vari-able (r.v.s) into the statistical literature. Negative association of r.v.s was introduced by Joag-Dev and Proschan[2]. Both positively associated samples and negatively associated samples are called associated samples. There has been a significant number of research papers on this field. Because of a wide range of applications of associated samples, there are many important results under associated samples.The empirical likelihood method to construct confidence intervals, proposed by Owen[3,4] in1988, has many advantages over its counterparts like the bootstrap method.The joint asymptotic distributions of estimators of a distribution function at a fi-nite number of different points under associated samples are studied. It is shown that the joint asymptotic distributions are multi-normal distributions. We use the blockwise technology[5] into empirical likelihood method so as to prove the logarithm empirical likeli-hood ratio of distribution functions at a finite number of different points is asymptotic the chi-square distribution under negatively associated samples, and then construct empiri-cal likelihood confidence regions for distribution functions at a finite number of different points.We also do a simulation study. The results show that the coverage probability of the empirical likelihood confidence interval of the distribution difference at any two points is quite good.Here we summary some new findings in theory:1. This paper extends the method of kernel estimation from a single point to multiple points. We investigate the joint asymptotic distributions of estimators of a distribution function at a finite number of different points under associated samples. It is shown that the joint asymptotic distributions are multi-normal distributions.2. This paper extends the empirical likelihood for distribution function from a single point to multiple points, by using the blockwise technology to prove that the asymptotic distribution of distribution functions at a finite number of different points has chi-square distribution under negatively associated samples, and constructs the joint empirical like-lihood confidence regions for a finite number of distribution functions.