Large Sample Properties For Several Statistical Regression Models Under Dependent Errors

Regression models have essential applications in many realistic fields.Until now many important and applicative models have been proposed.However,investigations for these regression models under dependent cases are not completed.Therefore,this work mainly discusses the large sample properties for the estimators in several statistical regression models under dependent errors.Firstly,consider the following multiple linear regression model:yi=xiT?+?i,i=1,...,n,n?1,where xi=(xi1,...,xid)T,1?i?n,are known design vectors,y1,…,yn are observed values,?1,?2;…,?n are random errors with zero mean,and?=(?1,…,?d)T are unknown parameters and should be estimated.Under appropriate conditions,we establish the weak consistency and complete consistency for the least squares estimator of unknown parameter ? under m-END random errors.The results complement and extend the corresponding ones of Hu et al.[6]and Yang et al.[7].Next,consider the nonparametric regression model as follows:Yni=g(xni)+?ni,i=1,…n,n?1,where g(x)is an unknown function defined on Rd,d?1,xni are known d dimensional vectors,?ni are random errors with zero mean.For each n?1,the joint distribution of(?n1,?n2,…,?nn)is the same as that of(?1,?2,…,?n).Under END random errors,we first establish the results on strong consistency,complete consistency and mean consistency for the P-C estimator,which improve and extend the corresponding results of Priestley and chao[8]as well as Yang and Wang[10];Then,we investigate the asymptotic normality and Berry-Esseen type bound for general weighted estimator under ANA random errors.The result reveals that under suitable conditions,the Berry-Esseen type bound for general weighted estimator under ANA random errors can also achieve the optimal rate obtained in Yang[127]under NA random errors;At last,we further obtain the complete consistency for the general weighted estimator under m-ANA random errors.Whereafter,we consider the following partially linear regression model:yi=xiT?+g(ti)+Vi,1?i?n,where xi an ti are design points,? is an unknown parameter to be estimated,g(·)is an unknown function defined on[0,1],yi are observed values,Vi are random errors with zero mean.With the errors formed by a linear process of ?-mixing innovations,we investigate the asymptotic normality and Berry-Esseen type bound for the estimators in this semiparametric regression model,which extend the corresponding results in Liang and Fan[128].We further consider the following more general partially linear regression model:Y(j)(xin,lin)=lin?+g(xin)+e(j)(xin),1?j?k,1?i?n,where tin?R and xin ? Rd and Xin ? Rd are known to be nonrandom,? is an unknown parameter to be estimated,g is an unknown continuous function on a compact set A in Rd,e(i)(xin)are immeasurable random errors,Y(i)(xin,tin)represent the j-th response variables which are observable at points xin,tin.Under m-END errors,we study the strong consistency,complete consistency and r-th mean consistency for the least squares estimator of parameter and weighted estimator of nonparametric part,which markedly improve and extend the corresponding results in References[33]-[37].We also investigate the following simple linear errors-in-variables regression model:?i=?+?xi+?i,?i=xi+?i,1?i?n,where ?,?,x1,x2,…are unknown constants(parameters),(?1,?1),(?2,?2),are random vectors and ?i,?i,i=1,2,...are observable.We first establish the complete convergence result for weighted sums of WOD random variables under some very general conditions,the dominating coefficients are only required to increase polynomially and the moment condition has no relation to the dominating coefficients.Based on this result,we further obtain some general results on the rate of complete consistency for the least squares estimators in simple linear errors-in-variables regression model under WOD errors,which can also derive complete consistency under some weaker conditionsFinally,we consider the following heteroscedastic partially linear errors-in-variables model yi=?i?+g(ti)+?i,xi=?i+?i,where ?i =?iei,?i2=f(ui),(?i,ti,ui)are design points,(ti,xi,yi)are observed samples,?i are the potential variables which are unobservable,yi are the response variables,and xi are observed with measurement errors ?i with E?i=0,ei are random errors with Eei=0.? ? is an unknown slope parameter.f(·)and g(·)are unknown functions defined on closed interval[0,1]Some general results on the rates of strong consistency for the least squares estimators and weighted least squares estimators in heteroscedastic partially linear errors-in-variables model for WOD random variables are presented The errors {ei,i?1} and {?i,i?1} can be both WOD with polynomially increasing dominating coefficients,while the moment condition has no relation to the dominating coefficients.The results markedly improve and extend the corresponding ones of Zhang and Wang[57].