Uniform Convergence Rate of Estimators of Autocovariances in Partly Linear Regression Models with Correlated Errors

Consider the partly linear regression model \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ y_{i} = {x}'_{i} \beta + g{\left( {t_{i} } \right)} + \varepsilon _{i} ,\;\;{\kern 1pt} 1 \leqslant i \leqslant n $$\end{document}, where yi’s are responses, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ x_{i} = {\left( {x_{{i1}} ,x_{{i2}} , \cdots ,x_{{ip}} } \right)}^{\prime } \;\;\;{\text{and}}\;\;\;t_{i} \in {\cal T} $$\end{document}are known and nonrandom design points, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\cal T} $$\end{document} is a compact set in the real line \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\cal R} $$\end{document}, β = (β1, ··· , βp)' is an unknown parameter vector, g(·) is an unknown function and {εi} is a linear process, i.e.,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \varepsilon _{i} {\kern 1pt} = {\kern 1pt} {\sum\limits_{j = 0}^\infty {\psi _{j} e_{{i - j}} ,{\kern 1pt} \;\psi _{0} {\kern 1pt} = {\kern 1pt} 1,\;{\kern 1pt} {\sum\limits_{j = 0}^\infty {{\left| {\psi _{j} } \right|} < \infty } }} } $$\end{document} , where ej are i.i.d. random variables with zero mean and variance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sigma ^{2}_{e} $$\end{document}. Drawing upon B-spline estimation of g(·) and least squares estimation of β, we construct estimators of the autocovariances of {εi}. The uniform strong convergence rate of these estimators to their true values is then established. These results not only are a compensation for those of [23], but also have some application in modeling error structure. When the errors {εi} are an ARMA process, our result can be used to develop a consistent procedure for determining the order of the ARMA process and identifying the non-zero coeffcients of the process. Moreover, our result can be used to construct the asymptotically effcient estimators for parameters in the ARMA error process.