Empirical Likelihood Inference For Population Quantiles And M-functionals Under Negatively Associated Sample

The concept of negative association was introduced and studied by Block et al.[Some con-cepts of negative dependence, Ann. Probab.10(1982),765-772] and Joag-Dev and Proschan [Negative association of random variables with applications, Ann. Statist.11(1983),286-295]. The convergence of the sums of negatively associated random variables, because of their wide applications, were studied extensively. The empirical likelihood (EL) method to construct con-fidence intervals, proposed by Owen [Empirical likelihood ratio confidence intervals for a sin-gle functional, Biometrika.75(1988),237-249; Empirical likelihood ratio confidence regions, Ann. Statist.18(1990),90-120], has many advantages over its counterparts like the normal-approximation-based method and the bootstrap method.This paper mainly studies empirical likelihood inference for population quantiles and M-functionals under NA samples, and uses the blockwise technology into empirical likelihood method, so as to prove the parameters of the logarithm empirical likelihood ratio is asymptotic the chi-square distribution, so we constructed empirical likelihood confidence interval for popula-tion quantiles and M-functionals in the absence of auxiliary information under NA samples, and constructed empirical likelihood confidence interval of M-functionals in the presence of auxiliary information under NA sample. We also do a simulation study, the results show that empirical likeli-hood confidence interval of population quantiles and M-functionals has good coverage probability under small samples. At the same time, coverage probability gradually close to name confidence level with the size of sample, and the length of interval decreased with the increase of the sample size; The coverage probability of the Empirical likelihood confidence interval of M-functionals in the presence of auxiliary information is more close to name confidence level than the Empirical likelihood confidence interval of M-functionals in the absence of auxiliary information, and the length of interval is a little bit small. Here we summary some new findings in this paper:1、 We use blockwise technology into empirical likelihood method. In the article, we accord-ing to blockwise technology of Kitamura (1997) to prove the logarithm of empirical likelihood ratio of population quantiles and M-functionals is asymptotic the chi-square distribution, and con-struct the empirical likelihood confidence interval.2、This paper we study empirical likelihood inference for M-functionals under NA samples, and use auxiliary information to improve empirical likelihood, contain the logarithm of empirical likelihood ratio about M-functionals in the absence of auxiliary information and in the presence of auxiliary information, and by using the blockwise technology to prove it is asymptotic the chi-square distribution, and construct the empirical likelihood confidence interval of M-functionals. Through the simulation results show that, the empirical likelihood confidence interval coverage probability of M-functionals in the presence of auxiliary information is more close to name con-fidence level than in the absence of auxiliary information, and interval length is more smaller.