Acta Mathematicae Applicatae Sinica, English Series

, Volume 19, Issue 3, pp 363–370 | Cite as

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

Original papers
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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 

2000 MR Subject Classification

62J05 62E20 

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Copyright information

© 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

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