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

In1967, Esary et al.introduced the concept of positively associated random vari-ables (r.v.s) into the statistical literature. Negative association of r.v.s was introduced by Joag-Dev and Proschan in1983. Both positively associated samples and negatively associated samples are called associated samples. It has been discussed by many statis-ticians. As the sums of associated random variables sequence is not only widely used in some reliability theory problems, hydromechanics, time series models, but also used in ecological system and some clinical trials.The statistical inference of density functions at a finite number of different points under associated samples are studied. It is shown that the joint asymptotic distributions are multi-normal distributions by blockwise technology. We extend the method in Qin et al.and Xiong and Lin of kernel estimation from a single point to multiple points so as to prove the logarithm empirical likelihood ratio of density functions at a finite number of different points is asymptotic the chi-square distribution under associated samples, and then construct empirical likelihood confidence regions for density 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 density functions difference at any two points is quite good.The new findings in this paper may be summarized as follows:1. This paper investigates the joint asymptotic distributions of estimators of a density function at a finite number of different points under associated samples, by using the blockwise technology to prove that the joint asymptotic distributions are multi-normal distributions.2. This paper extends the empirical likelihood for density function from a single point to multiple points, It is shown that the asymptotic distribution of density functions at a finite number of different points has chi-square distribution under associated samples and constructs the joint empirical likelihood confidence regions for a finite number of density functions. 3. This paper constructs the empirical likelihood confidence interval of the density functions difference at any two points.