Empirical Likehood For Nonparametric Regression Functions Under Negatively Associated Samples

The concept of negative association was introduced and studied by Block et al.[Some concepts 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]. Since the concept of NA sequences is given, lots of scholars studied the nature of the limits of NA sequences (such as the central limit theorem, strong law of large numbers and complete convergence, etc.). As the sums of NA random variables sequence ia not only widely used in system reliability theory, percolation theory and multivariate statistics, but in engineering such as communication, system, meteorological sciences, geological statistics and in graphic analysis, risk analysis, marine sci-ence, ecology, etc., domestic and international statisticians give rise to the widespread concern of it. Many specific statistical problems under NA dependent samples also began to study.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 independent data and the usual EL cannot be used properly for dependent data. Consider the following nonparametric regression model where Y is a scalar response variable, X∈Rd is a vector of random design variable. 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. Qin et al.[Confidence intervals for nonparametric regression functions under negatively associated errors, J. Nonparametr. Stat.20(2011),645-659] studied on fixed design situation the construction intervals for a nonparametric regression function under a negatively associated sample by using the blockwise EL method, while on random design sit-uation there is no literature studied the EL for a nonparametric regression function. This paper, as a promotion of the conclusion of this paper, by using the blockwise EL method, studied on random design situation the construction intervals for a nonparametric regression function under a negatively associated sample. It is shown that under some condition the (-2times) logarithm of blockwise EL ratio statistic is asymptotically χ2-type distributed, and we construct the EL confi-dence regions for θ=m(x).The new findings in this paper may be summarized as follows:(1) This is the first time when the EL confidence regions of nonparametric regression model with random designs under NA data are constructed.(2) The method in this paper suggests a way to construct EL confidence regions for nonpara-metric regression model under more general dependent conditions.