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

Uniform Convergence Rate of Estimators of Autocovariances in Partly Linear Regression Models with... 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 = (β 1, ··· , 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

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

, Volume 19 (3) – Mar 3, 2017

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

Acta Mathematicae Applicatae Sinica, English Series Vol. 19, No. 3 (2003) 363–370 Uniform Convergence Rate of Estimators of Autocovariances in Partly Linear Regression Models with Correlated Errors 1 2 3 4 Jin-hong You , Gemai Chen ,Min Chen ,Xue-leiJiang University of Regina, Regina, Saskatchewan, S4S 0A2, Canada University of Calgary, Calgary, Alberta, T2N 1N4, Canada (E-mail: gchen@math.ucalgary.ca) 3,4 Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, China 3 4 (E-mail: mchen@amss.ac.cn; jxl5@amss.ac.cn) Abstract Consider the partly linear regression model y = x β + g(t )+ ε , 1 ≤ i ≤ n,where y ’s are i i i i responses, x =(x ,x ,··· ,x ) and t ∈T are known and nonrandom design points, T is a compact set i i1 i2 ip i in the real line R, β =(β ,··· ,β ) is an unknown parameter vector, g(·) is an unknown function and {ε } is 1 p i ∞ ∞ a linear process, i.e., ε = ψ e ,ψ =1, |ψ | < ∞,where e are i.i.d. random variables with zero i j i−j 0 j j j=0 j=0 mean and variance σ . Drawing upon B-spline estimation of g(·) and least squares estimation of β,we construct estimators of the autocovariances of {ε }. 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 } 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 coeﬃcients of the process. Moreover, our result can be used to construct the asymptotically eﬃcient estimators for parameters in the ARMA error process. Keywords Uniform strong convergence rate, autocovariance and autocorrelation, B-spline estimation,...

/lp/springer_journal/uniform-convergence-rate-of-estimators-of-autocovariances-in-partly-CL99G0nmwL
Publisher
Springer Journals
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
D.O.I.
10.1007/s10255-003-0111-5
Publisher site
See Article on Publisher Site

### 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 = (β 1, ··· , 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.

### Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Mar 3, 2017

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