Acta Mathematicae Applicatae Sinica, English Series

, Volume 19, Issue 3, pp 363–370

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

• Jin-hong You
• Gemai Chen
• Min Chen
• Xue-lei Jiang
Original papers

## Abstract

Consider the partly linear regression model $$y_{i} = {x}'_{i} \beta + g{\left( {t_{i} } \right)} + \varepsilon _{i} ,\;\;{\kern 1pt} 1 \leqslant i \leqslant n$$, where y i ’s are responses, $$x_{i} = {\left( {x_{{i1}} ,x_{{i2}} , \cdots ,x_{{ip}} } \right)}^{\prime } \;\;\;{\text{and}}\;\;\;t_{i} \in {\cal T}$$are known and nonrandom design points, $${\cal T}$$ is a compact set in the real line $${\cal R}$$, β = (β1, ··· , β p )' is an unknown parameter vector, g(·) is an unknown function and {ε i } is a linear process, i.e.,$$\varepsilon _{i} {\kern 1pt} = {\kern 1pt} {\sum\limits_{j = 0}^\infty {\psi _{j} e_{{i - j}} ,{\kern 1pt} \;\psi _{0} {\kern 1pt} = {\kern 1pt} 1,\;{\kern 1pt} {\sum\limits_{j = 0}^\infty {{\left| {\psi _{j} } \right|} < \infty } }} }$$ , where e j are i.i.d. random variables with zero mean and variance $$\sigma ^{2}_{e}$$. Drawing upon B-spline estimation of g(·) and least squares estimation of β, we construct estimators of the autocovariances of {ε i }. 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 {ε i } 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.

## Keywords

Uniform strong convergence rate autocovariance and autocorrelation B-spline estimation correlated error partly linear regression model

62J05 62E20

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© Springer-Verlag Berlin Heidelberg 2003

## Authors and Affiliations

• Jin-hong You
• 1
• Gemai Chen
• 2
• Min Chen
• 3
• Xue-lei Jiang
• 3
1. 1.University of ReginaReginaCanada
2. 2.University of CalgaryCalgaryCanada
3. 3.Academy of Mathematics and System SciencesChinese Academy of SciencesBeijingChina