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

The φ-mixing condition was firstly introduced and studied by Ibragimov [Some limit theo-rems for stochastic processes stationary in the strict sense, Dokl. Akad. Nauk SSSR.125(1959):711-714.]. As measures of weak dependence, the concept of0-mixing has been used extensively in the time series literature. As its wide applications,the convergence of the sums of φ-mixing ran-dom variables were studied extensively.Bradley [Basic properties of strong mixing conditions:a survey and some open questions. Probability surveys.2(2005),107-144] made a good survey on the φ-mixing condition as well as other commonly used mixing conditions. In order to constructing confidence intervals of interested parameters,Owen[Empirical likelihood ratio confidence intervals for a single functional. Biometrika,75(1988):237-249.] proposed the EL method,which has many advantages over its counterparts,like the normal-approximation-based method.This paper mainly firstly studies the joint asymptotic distributions of kernel estimators of probability density function at a finite number of points under0-mixing sample, and then discusses the construction the confidence region by empirical likelihood method. It shows that the joint asymptotic distribution is proved to be asymptotic normal distribution. Then, as an application of this result,one can obtain the asymptotic distribution of the estimator of the difference of at any two probability density functions.We use the blockwise technology into empirical likelihood method to prove the parameters of logarithm empirical likelihood (EL) ratio statistic is asymptotically X2-type distribution,which is used to obtain the EL-based confidence region.The new findings in this paper may be summarized as follows:(1)Li Junyun in her Master’s thesis studied the construction of confidence interval for proba-bility density function under φ-mixing samples.This paper generalizes the results to a finite number of points,expanding the scope of the empirical likelihood method.(2)In this paper,we prove that the joint asymptotic distribution is asymptotic normal distribu-tion. As an application of this result, the asymptotic distribution of the estimator of the difference of at any two probability density functions is also obtained.(3)The method used in this paper suggests a way to construct EL confidence regions under more general mixing conditions.