Confidence Intervals For Nonparametric Regression Functions Under φ-mixing Samples

Likelihood is one of the most important statistical methods.The EL method to construct confidence 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 (e.g., Hall P. and La S. B.[Methodology and algorithms of empirical likelihood, Internat Statist. Rev.58(1990),109-127] or Hall P.[The Bootstrap and Edgeworth Expansion, New York:Springer-Verlag,1992]). Owen[Empirical like-lihood for linear models, Ann. Statist.19(1991),1725-1747] used the EL method to construct confidence intervals for the vector of regression parameters in a linear model.It should be noted that the above work using EL method seems to focus on independent data and the usual EL can not be properly for dependent data.The Φ-mixing condition was introduced by Ibragimov [Some limit theorems for stochastic processes stationary in the strict sense, Dokl. Akad. Nauk SSSR.125(1959),711-714], and was also studied by Cogburn [Asymptotic properties of stationary sequences, Univ. Calif. Publ. Statist.3(1960),99-146]. The concept of Φ-mixing has been used extensively in the time series literature as measures of weak dependence.Bradley[Basic properties of strong mixing conditions:a survey and some open questions, Probab. Surveys.2(2005),107-144] made a good survey on the Φ-mixing condition as well as other commonly used mixing conditions.In this paper, we study the construction of confidence intervals for a nonparametric regression function undera Φ-mixing sample by using the blockwise technique. It is shown that the blockwise empirical likelihood (EL) ratio statistic is asymptotically x2distributed. The result is used to obtain EL based confidence intervals for a nonparametric regression function. Results of a simulation study on the finite sample performance of the proposed confidence intervals are reported.There is a lack of research on the construction of the EL-based confidence intervals for a nonparametric regression function under a0-mixing observations.lt is possible to generate the results in this paper to broader mixing condition,which is left for our future study.We also note that the regularity conditions in a broader mixing setting would be much more complex and could not be expected as nice as those in the Φ-mixing situation.