Empirical Likelihood For Linear Models With Random Designs Under Negatigvely Associated Samples

The concept of negative association was introduced and studied by Block et al.[Some con-cepts of negative dependence, Ann. Probab.10(1982),765-772] and Joag-Dev and Proschan [Negative association of random variables with applications, Ann. Statist.11(1983),286-295]. The convergence of the sums of negatively associated random variables, because of their wide applications, were studied extensively.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(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 indepen-dent data and the usual EL cannot be used properly for dependent data.Consider the following linear regression model Y=Xτβ+ε, where Y is a scalar response variable, X∈Rr is a vector of random design variable,β∈Rr is a vector of regression parameters, and error ε€R satisfies E(ε|X)=0. Let X1,…, Xn be the observations of design vector, Y1,…, Yn be the corresponding observations of Y. We assume that {X1, Y1,X2, Y2,^,Xn,Yn} is a NA random variable sequence. In this paper, by using the blockwise EL method, we construct the EL confidence regions for the regression vector β in this linear model with0-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 with random designs under NA 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 dependent conditions.