Theory Research On Asymptotics Of Estimators Of Nonparametric Regression Function

Let(X,Y)be a R^d×R valued random vector and (X₁,Y₁),(X₂,Y₂),...,(X_n,Y_n)be a random sample drawn from(X,Y).If E|Y|＜∞,the regression function of Y given X is defined as m(x)=E(Y|X=x),x∈R^d.How to estimate m(x)from the sample {(X,Y),1≤i≤n} has been one of the most significant things in probabi -lity and statistics.Nadaraya-Waston suggested to use kernel estimatior m_1n(x).Then the famous statistician professor Cheng ping in China proposed modified kernel estimator(?)_1n(x)and(?)_2n(x)of regression function given above.Paul Algoet and LászlóGy(?)rfi proposed the estimator m_4n(x)based on partitioning of regression function.Then professor Zhao Ling Cheng(2002)modified estimator m_4n(x)and constructed modified estimator(?)_4n(x)based on partitioning by the way of censoring.According to the estimators of regression function given above,a lot of researchers have studied their large sample properties under many kinds of samples, such as convergence,convergence rate,asymptotic normality e.t.Professor Lin'g Nengxiang(2004)proved strong convergence and its rate of modified partitioning estimation under dependence samples,(2006) proved the asymptotic normality of partitioning and modified partitioning estimation function.Stimulated by this,I proposed asymptotic normality of partitioning estimatior under dependence sample,modified partitioning recursive estimator and proved its strong consistency under geometricα—mixing sample and asymptotic normality under i.i.d sample.Research shows that asymptotic normality of estimator(?)_2n(x) underα—mixing sample with the beter weakly mixing dependent condition haven't been researched,yet.While this large sample property plays an important role in the theories of nonparametric regression estimation.Therefore it comes to be the fourth problem studied in this paper.