Simultaneous Statistical Inference Of Distribution Function At A Finite Number Of Points Under Φ-mixing Sample

The concept of0-mixing condition was first introduced and researched by Ibragimov,and was also researched by Cogburn,The concept of (?)-mixing has been employed extensively in the time series literature as measures of weak dependence.Bradley made a very good review on the (?)-mixing condition as well as other commonly employed mixing conditions.Due to its widely ap-plications,the convergence of the sums of (?)-mixing random variables were applied extensively by Utev(1990),Chen(1991),Herrndorf(1983),Peligrad(1985),Sen(1971,1974),Shao(1993)å'ŒWang et al(2009).To construct confident intervals by using the method of empirical likelihood (EL) was first proposed by Owen(1988).After a lot of research conclusion,the EL method has many advantages compared with other common methods of statistical inference.such as,domain properties.transform invariance and so on.And the shape of the Confidence regions is completely determined by the sample data completely,not need tectonic axis statistics.further,Owen(1990) was also constructed confident regions in distributed random vector under independent identically distributed.and it should be noted that the usual EL method can be onle used properly for independent data, but not for dependent data.Kitamura(1997)first proposed the method of blockwise empirical likelihood to constuct con-fidence intervals for parameters under mixing samples.Chen and Wong (2009)has constuct confi-dence intervals for quantiles under (?)-mixing samples in the same way as above.The kernel esti-mation principle of probability density function was first putting forwarded by Rosenblatt, similar to the kernel estimation principle of distribution function.Here we mainly use the blockwise EL method to construct the joint asymptotic distribution and its empirical likelihood ratio statistics of the kernel estimator of distribution function in a finite number of points under (?)-mixing sam-ples,The results showed that the joint asymptotic distribution is Normal distribution,the asymptotic distribution of blockwise empirical likelihood ratio statistics is chi-square distribution.finally,the performances of the EL-based confidence regions and the normal-approximation-based confidence regions were compared through simulation studies.This paper mainly discusses the following aspects: (1) In chapter l,we mainly introduces general situation the research and development of (?)-mixing sequence.the research and development process and current situation of the empirical like-lihood.the research process of the kernel estimator of distribution function.(2) In chapter2,we mainly study the joint asymptotic distribution of the kernel estimator of distribution function in a finite number of points under (?)-mixing samples by the method of blockwise.(3) In chapter3,combined with the conclusion of asymptotically normal in chapter2,further evidence that its empirical likelihood ratio statistics of the kernel estimator of distribution function in a finite number of points by EL method under (?)-mixing samples.Uniqueness of this paper is mainly manifested in:(1) This article is the first time constructed the joint asymptotic distribution of the kernel esti-mator of distribution function in a finite number of points under0-mixing samples by the method of blockwise.and further proved that empirical likelihood ratio statistics of the kernel estimator of distribution function in a finite number of points by EL method under0-mixing samples.eventually work out empirical likelihood confidence regions.(2) The research methods and results of this paper can be used for reference to construct the joint asymptotic distribution of the kernel estimator of distribution function and the empirical likelihood ratio statistics of the kernel estimator of distribution function in a finite number of points.