Empirical Likelihood Confidence Intervals For Probability Density Functions In Presence Of Auxiliary Information Under Associated Samples

In this article, we mainly investigate the problem of the empirical likelihood (EL) confidence intervals for probability density functions in presence of auxiliary information under negatively associated(NA) samples and strong mixing samples. Under auxiliary information conditions, the asymptotic distribution of the empirical likelihood ratio statistic for probability density functions is obtained. This article includes three chap-ters:The first chapter is introduction. The concepts and research progress of NA samples and strong mixing samples are reviewed, and the research progress of EL method is also reviewed. We also describe the structure and contents of this thesis in this chapter.The second chapter studies the EL confidence intervals for probability density func-tions in presence of auxiliary information under negatively associated(NA) samples. We present the specific form of the likelihood ratio statistic by using the block-wise tech-nique and discuss its asymptotic distribution. The asymptotic distribution of the EL ratio statistic for probability density functions in presence of auxiliary information is proved to be x2-distribution finally. The result is used to obtain EL based confidence intervals on the probability density.The third chapter studies the EL confidence intervals for probability density func-tions in presence of auxiliary information under strong mixing samples. Similarly to the case of NA samples, we have proved that the asymptotic distribution of the EL ratio statistic for probability density functions in presence of auxiliary information un-der strong mixing samples isx2 -distribution.The result is used to obtain EL based confidence intervals on the probability density.The innovation of this thesis is mainly reflected in the following three aspects:1.We obtain the block EL ratio statistic in presence of auxiliary information by using the block EL method, and the asymptotic distribution of the block EL ratio statistic is proved.2.We obtain EL based confidence intervals on the probability density in presence of auxiliary information by using the asymptotic distribution of the EL ratio statistic for probability density functions.3.A hypothesis testing problem is considered under NA samples, and it is shown that the asymptotic efficiency of the test is not changed in presence of auxiliary information.