Statistical Inference For M-functionals In The Presence Of Auxiliary Information Under α-mixing Samples

This paper studies the M-functional statistical inference under a-Mixing sample in the pres-ence of auxiliary information. First of all, in this paper,a-mixing sequence, the M-Functional and a simple empirical likelihood method are introduced, enables us to have a preliminary understand-ing about the research objects and methods. In addition, create a new M-estimator with additional information by using the empirical likelihood method, which proves the asymptotic distribution of M-estimate under a-Mixing sample in the presence of auxiliary information is normal distribution. Finally, prove that the limit distributions of empirical log-likelihood ratio statistic for M-functional after using group empirical likelihood is chi-square distribution, so empirical likelihood confidence interval for M-functional is constructed.The innovation of this article is mainly manifested in:1. This paper constructed the limit distributions of empirical log-likelihood ratio statistic for M-functional under a-Mixing sample in the presence of auxiliary information is chi-square distribution and empirical likelihood confidence interval which enlarge the application scope of empirical likelihood for the first time.2. This article proved the asymptotic distribution of M-estimate under a-Mixing sample in the presence of auxiliary information is normal distribution clearly, extended M-estimate under independent sample in the presence of auxiliary information.