Asymptotic Properties Of Estimators In Linear And Partial Linear Regression Models Under WOD Errors

Regression model is a statistical model with early development,rich theory and strong applicability in statistics.Then,due to the need of practical appli-cation,the regression model has been developing continuously.And it has been developed from the initial parameter regression to the non-parametric regres-sion model and the semi-parametric regression model.This thesis mainly dis-cusses linear regression model and partial linear regression model,which are the classical representatives of parametric regression model and semi-parametric regression model respectively.In the regression models,the usual assumption that the random errors are independent is not plausible,especially for continuous collected economic data.Therefore,the thesis mainly considers a relatively broad dependent random error:WOD random error,and linear process error generated by WOD random variables.First,consider the classical linear model as follows:Yi?xi'?+ei,i=1,…,n,n? 1,(1)where x1,x2,...·,xn are p×1 known design vectors,e1,e2,...,en are mean zero WOD random errors,and ? is a p x 1 unknown parameter vector.By using truncation method of random variables and inequalities,the almost sure convergence for weighted sums of WOD random variables is established.Based on the strong convergence that we established and the Bernstein-type inequal-ity,we investigate the strong consistency of M estimators of ? in the linear model(1)with WOD random errors,which extends the corresponding ones of Chen and Zhao[113][114]and Wu and Jiang[65].Second,on the basis of the above result,we further study the strong consistency of M estimators.By the strong laws of large numbers for WOD random variables obtained by Wang and Cheng[90]and Chen et al.[115],we establish a border strong convergence for weighted sums of WOD random variables,and apply it to investigate the strong consistency of M estimators.Compared with Wu and Jing[65],the result is also true without the restriction of moment condition when ?=1.In addtion,it weakens the assumptions,and extends the result of Wang and Hu[116]for NSD random errors.Third,it is well known that Bernstein-type inequality is the important tool of probability and mathematical statistics,but many applications of these inequalities are restricted to the bounded condition.For this reason,we es-tablish an exponential inequality for WOD random variables without bounded condition,and its proof line is different from that of the classical Bernstein-type inequality.The result extends the corresponding one in Chen and Sung[117]for NOD sequences.Next,we apply the established inequality to study the strong linear representation of M estimators ?n of ? in the linear regression model(1).The obtained result extends the corresponding one in Rao and Zhao[118]for independent random errors,which gave no detailed proofs.For dependent random errors,it is also true for NA,NSD,NOD and END random errors.Forth,consider the partial linear model I as follows:yi=xi?+g(ti)+?iei,i=1,2?...,n,(2)where ?i2=f(ui),(xi,ti,vi)are known fixed design points,? is an unknown parameter to be estimated,g(·)and f(·)are unknown functions defined on compact set A(?)R,ei are mean zero WOD random errors,which is stochasti-cally dominated by a random variable e.The moment consistency and strong consistency for least squares estimators and weighted least squares estimators of ? and g in the partial linear model(2)under some more mild conditions are studied.These results extend and improve the corresponding ones of Zhou and Hu[29]and Baek and Liang[27]respectively.Last,consider the partial linear model II as follows:yi(n)=xi(n)?+g(ti(n))+?i(n),i=1,2,...,n,?1,(3)where g is an unknown function defined on a compact set A C Rp,and ? is an unknown parameter in R,xi(n)and ti(n)are known to be nonrandom,yi(n)represents the i-th response which is observable at points xi(n)and ti(n),?i(n)is a random error.Assume that for eacn n,(?1(n),?2(n),...,?n(n)has the same distribution as that of(?1,?2,...,?n),where ?i is of the following form:(?)where {ei} is a sequence of identically distributed WOD random variables with Ee0=0,and {?j} is a sequence of real numbers satisfying(?)|?j|<?.·In the thesis,we first prove the complete convergence of linear process {?i},and then provide the complete consistency for the least squares estimators of? and g(·)by using the complete convergence result that we established.