Empirical Likelihood For Linear Models Underφ-mixing Samples

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.The empirical likelihood (EL) method to construct confidence intervals, proposed by Owen [Empirical likelihood ratio confidence intervals for a single 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. Owen [Empirical likelihood for linear models, Ann. Statist.19,1725-1747.] 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 cannot be used properly for dependent data.Consider the following linear regression model where y is a scalar response variable, x∈Rr is a vector of fixed design variable,β∈Rr is a vector of regression parameters, and error (?)∈R is a random variable. Let x1,…, xn be the design vectors, y1,…, yn be the corresponding observations, and∈1,…,∈n be the random errors. We assume that{∈j, j>1} is a φ-mixing random variable sequence. We note that y1,…, yn is a0-mixing sample in this case. In this paper, by using the blockwise EL method, we construct the EL confidence regions for the regression vector β in this linear model with φ-mixing errors. Through simulation studies, we compare the performances of the EL-based confidence regions and the normal-approximation-based confidence regions.The new findings in this paper may be summarized as follows:(1) This is the first time when the EL confidence regions of regression vector in a linear model under φ-mixing data are constructed. (2) The method in this paper suggests a way to construct EL confidence regions for regression vector in a linear model under more general mixing conditions, which is left for our future study.