Asymptotic Properties Of The Estimators In Nonparametric And Partially Linear Regression Models With Mixing Errors

Regression analysis is a statistical tool used for determining the relationships between ran-dom variables.When using a regression model,the researchers seek to ascertain the causal effects of random variables upon each other.In order to explore such issues,the researcher assembles the effects of the explanatory variable upon the dependent variables.In this thesis,three different regression models are being discussed:Nonparametric regression mod-el,partially linear regression models and heteroscedastic partially linear regression model Therefore,this thesis focuses on the asymptotic properties of the estimators of the nonpara-metric and partially linear linear regression models under dependent errors.Three problems relevant with the above mentioned subjects are consideredFirst,we investigate the effect of dependent errors in the fixed design nonparametric regression model.Under some mild conditions,we obtain the complete consistency and asymptotic normality for the weighted estimator in the fixed design nonparametric regres-sion models.In addition,a simulation study is undertaken to investigate finite sample behavior of the estimator and a real data application of our estimator is then presentedNext,we study the consistency properties for the partially linear regression model Y(i)(xin,tin)=tin?+g(xin)+e(j)(xin),1?j?k,1?i?n,where xin?Rp,and tin ? R are known to be nonrandom,g(·)is an unknown continuous function on a com-pact set A in Rp,e(j)(xin)are(?,?)-mixing random errors with mean zero,Y(i)(xin,tin)are random variables which are observable at points xin and tin.By using the probability inequalities and moment inequalities,we obtain the strong consistency,complete consistency and mean consistency for the estimators ?k,n and 9k,n of ? and g,respectively.Simulation studies are conducted to demonstrate the performance of the proposed procedure and a real example is analysed for illustrationFinally,we investigate the estimators for the heteroscadastic partially linear regression model under dependent errors defined by yi=xi?+g(ti)+?i(1?i?n),where ?i=?iei,?i2=f(ui),the design points(xi,ti,ti)are known and nonrandom,? is an unknown pa-rameter to be estimated,the functions g(·)and f(·)are unknown,which are defined on closed interval[0,1],and the random errors {ei} are(?,?)-mixing random variables.When the model is heteroscedastic,the unknown parameter ?and the unknown function g(·)are approximated by the weighted least squares estimators.We derive the asymptotic normality of the weighted least squares estimators under some suitable conditions.Simulation studies are conducted to demonstrate the finite sample performances of the proposed procedure Finally,we use real data to examine the dependence between oil prices and exchange rates.