Consider the partly linear regression model % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D % aebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaq % Fr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qq % Q8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaca % WG5bWaaSbaaSqaaiaadMgaaeqaaOGaeyypa0JabmiEayaafaWaaSba % aSqaaiaadMgaaeqaaOGaeqOSdiMaey4kaSIaam4zamaabmaabaGaam % iDamaaBaaaleaacaWGPbaabeaaaOGaayjkaiaawMcaaiabgUcaRiab % ew7aLnaaBaaaleaacaWGPbaabeaakiaacYcacaaMe8UaaGjbVlaayI % W7caaIXaGaeyizImQaamyAaiabgsMiJkaad6gaaaa!53DA! $$ y_{i} = {x}''_{i} \beta + g{\left( {t_{i} } \right)} + \varepsilon _{i} ,\;\;{\kern 1pt} 1 \leqslant i \leqslant n $$ , where y i?s are responses, % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D % aebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaq % Fr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qq % Q8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaca % WG4bWaaSbaaSqaaiaadMgaaeqaaOGaeyypa0ZaaeWaaeaacaWG4bWa % aSbaaSqaaiaadMgacaaIXaaabeaakiaacYcacaWG4bWaaSbaaSqaai % aadMgacaaIYaaabeaakiaacYcacqWIVlctcaGGSaGaamiEamaaBaaa % leaacaWGPbGaamiCaaqabaaakiaawIcacaGLPaaadaahaaWcbeqaaO % Gamai4gkdiIcaacaaMe8UaaGjbVlaaysW7caqGHbGaaeOBaiaabsga % caaMe8UaaGjbVlaaysW7caWG0bWaaSbaaSqaaiaadMgaaeqaaOGaey % icI4Saamivaaaa!5D82! $$ x_{i} = {\left( {x_{{i1}} ,x_{{i2}} , \cdots ,x_{{ip}} } \right)}^{\prime } \;\;\;{\text{and}}\;\;\;t_{i} \in {\cal T} $$ are known and nonrandom design points, % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D % aebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaq % Fr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qq % Q8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaca % WGubaaaa!393B! $$ {\cal T} $$ is a compact set in the real line % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D % aebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaq % Fr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qq % Q8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaca % WGsbaaaa!3939! $$ {\cal R} $$ , ß = ( ß 1, ··· , ß p)? is an unknown parameter vector, g(·) is an unknown function and { e i} is a linear process, i.e., % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D % aebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaq % Fr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qq % Q8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacq % aH1oqzdaWgaaWcbaGaamyAaaqabaGccaaMi8Uaeyypa0JaaGjcVpaa % qahabaGaeqiYdK3aaSbaaSqaaiaadQgaaeqaaOGaamyzamaaBaaale % aacaWGPbGaeyOeI0IaamOAaaqabaGccaGGSaGaaGjcVlaaysW7cqaH % ipqEdaWgaaWcbaGaaGimaaqabaGccaaMi8Uaeyypa0JaaGjcVlaaig % dacaGGSaGaaGjbVlaayIW7daaeWbqaamaaemaabaGaeqiYdK3aaSba % aSqaaiaadQgaaeqaaaGccaGLhWUaayjcSdGaeyipaWJaeyOhIukale % aacaWGQbGaeyypa0JaaGimaaqaaiabg6HiLcqdcqGHris5aaWcbaGa % amOAaiabg2da9iaaicdaaeaacqGHEisPa0GaeyyeIuoaaaa!6ABD! $$ \varepsilon _{i} {\kern 1pt} = {\kern 1pt} {\sum\limits_{j = 0}^\infty {\psi _{j} e_{{i - j}} ,{\kern 1pt} \;\psi _{0} {\kern 1pt} = {\kern 1pt} 1,\;{\kern 1pt} {\sum\limits_{j = 0}^\infty {{\left| {\psi _{j} } \right|} < \infty } }} } $$ , where e j are i.i.d. random variables with zero mean and variance % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D % aebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaq % Fr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qq % Q8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacq % aHdpWCdaqhaaWcbaGaamyzaaqaaiaaikdaaaaaaa!3BF8! $$ \sigma ^{2}_{e} $$ . Drawing upon B-spline estimation of g(·) and least squares estimation of ß, 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. [ABSTRACT FROM AUTHOR]