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

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 {e 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 {e 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 {e 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.