Empirical Likelihood For Nonparametric Models Under Linear Process Errors

Linear process in time series analysis has a special important position and it works in hydrol-ogy, meteorology and areas of survival analysis, also it has a very wide range of applications in the economic, engineering and physics and other disciplines. In particular, limit theorems for charac-terization of linear processes in a variety of statistical inference for econometric models exported by limiting distribution of test statistics plays a vital role, there is considerable literature on the linear process limit to a deep and detailed studies.(e.g. Brockwell and Davis, Time Series:Theory and Methods[M]. New York:SpringerW,1987)Non-parametric regression model developed in the seventies of the twentieth century is an important statistical model. The regression function of non-parametric regression model can be any form, and fewer restrictions on the overall distribution, so it is broadly applied in practical problems and many scholars studied it, such as:Priestley and Chao (Non-parametric function fitting[J]. Journal of the Royal Statistical Society, Series B,1972,34:385-392)gives the Priestley-Chao type weight function of fixed design regression function estimates the condition of weak consistency, Benedetti (On the nonparametric estimation of regression functions[J]. J. Roy.Stat. Soc. B,1977,39(2):248-253)discussed the strong consistency of Priestley-Chao type weight function of fixed design regression function estimates estimate, Georgiev (Consistent nonpara-metric multiple regression:the fixed design case[J]. Journal of Multivariate Analysis,1988,25(1): 100-110) systematically studied the mean square and complete consistency of weight function estimates with fixed designs, Devroye (On the almost everywhere convergence of nonparametric regression estimates[J]. The Annals of Statistics,1981,9:1310-1319), Zhao Lincheng and Fang Zhaoben(strong consistency of nonparametric regression kernel estimation [J], Journal of Applied Mathematics,1985,8(3):268-276), Sun Dongchu (The strong consistency of regression func-tion Kernel estimation[J], The Annals of Mathematics,1985,6A(4):481-486)developed the large sample properties of kernel estimates in a nonparametric regression model with random designs. These are the properties of the estimation of the regression function under independent sample nonparametric.In recent years a large number of scholars to study on non-parametric modeling under linear process error, Tran et al.(Fixed-design regression for linear time series[J]. Ann. Statist,1996,24: 975-991)studied the estimation of regression function and obtained the asymptotic normality of the kernel estimators of regression function. Hu shuhe (Error of linear time sequence of regression model[J]. Mathematics annual,1999,20(6):733-740)extend and improve the results in the Tran et al.,also he got regression function of weight function estimates of asymptotic normality. Hu shuhe and Zhu chunhua (A Square Consistent Regression Model with Linear Time Series Error[J]. Journal of Anhui University,2000,24(3):18-22) obtain square consistency and uniform square consistency for regression model with linear time series error. Hu shuhe, Pan guangming and Gao qibing (Estimate problem of regression model with linear process errors [J]. University of applied mathematics journals,2003,18(1):81-90) for a nonlinear regression model and a linear model with linear process errors, the r-th mean,complete consistency and asymptotic normality for the estimators under suitable conditions are obtained.Specifically, Tran et al.(Fixed-design regression for linear time series[J]. Ann. Statist,1996, 24:975-991)studied the estimation of regression function and obtained the asymptotic normality of the kernel estimators of regression function. However, this result cannot be directly used to construct confidence intervals for regression function. In this paper, we study the construction of confidence intervals for a nonparametric regression function under linear process errors by using the blockwise technique. It is shown that the blockwise empirical likelihood (EL) ratio statistic is asymptoticallyχ2 distributed. The result is used to obtain EL based confidence intervals for the nonparametric regression function.The innovation of this paper is to construct empirical likelihood (EL) confidence intervals for regression function under linear process errors by using the blockwise technique. This is the issue that other literatures have not studied.