【作者】 姚梅；

【导师】 杜雪樵；

【作者基本信息】 合肥工业大学 ， 应用数学， 2006， 硕士

【摘要】 设（X₁,Y₁）,（X₂,Y₂）,…（X_n,Y_n）为从取值于R^d×R¹的总体（X,Y）中抽出的一个随机样本。若E|y|<∞,则称m（x）=E（Y|X=x）（x∈R^d）为Y关于X的回归函数。如何由上述样本对m（x）进行估计,一直是概率、统计界研究的热点之一。美国学者Paul Algoet和Laszlo Gyorfi（1999）提出了回归函数m（x）基于分割的估计m_n（x）;而后,我国著名统计学家赵林城教授（2002）对m_n（x）进行改良,并证明了在i.i.d样本下,改良基于分割估计的强相合性;在此基础上,凌能祥教授（2004）证明了在样本为同分布的φ混合序列时,回归函数改良基于分割估计的强相合性及收敛速度,（2005）证明了在样本为同分布的φ混合序列时,回归函数基于分割估计的强相合性。经研究我们发现,回归函数基于分割估计及其改良估计的其他大样本性质,国内外均无文献涉及,如混合相依较弱条件的α混合样本下估计量的强相合性及收敛速度;截尾数据下回归函数基于分割估计及其改良估计的渐近正态性等等,而这些性质在非参数回归估计理论中均占有重要的地位。 因此,本文主要对以下三个方面进行了研究（1）利用α混合序列的基本不等式,证明了同分布的α混合样本下回归函数基于分割估计的强相合性,积分绝对误差的强相合性与平均相合性;（2）利用α混合序列的Bemstein不等式,证明了同分布的α混合样本下回归函数改良基于分割估计的强相合性及收敛速度,积分绝对误差的强相合性与平均相合性;（3）利用截尾数据的一些性质和鞅的有关理论,在简洁合理的的条件下,证明了截尾样本下回归函数基于分割估计及其改良估计的渐近正态性。更多还原

【Abstract】 Let （X,Y） be a R^d×R¹ valued random vector and （X₁,Y₁）, （X₂,Y₂） …（X_n,Y_n） be a random sample drawn from （X,Y）. If E|Y| is finite, the regression function of Y given X is defined as m（x） = E（Y|X = x）, x∈R^d . How toestimate m（x） from the sample {（X₁,Y₁）,1≤i≤ n} has been one of the most significant things in probability and statistics.Professor Paul Algoet and Professor Laszlo Gyorfi （1999） in the U.S.A proposed partitioning estimate for regressionfunction m （x）;then the famous statistician Zhao Ling Cheng （2002） in China proposed the modified partitioning estimate for regression function m（x） and proved its strong consistency under i.i.d sample;based on this,Professor Ling Neng Xiang （2004） proved the strong consistency and convergence rate of modified partitioning estimate for regression function under sample that is identically distributed φ-mixing sequence;he （2005） proved the strong consistency ofpartitioning estimate for regression function under sample that is identically distributed <p - mixing sequence.By research,we know that there are many other large sample properties such as the strong consistency of partitioning estimate and modified partitioning estimate under a - mixing sample with the better weakly mixing dependent conditions;the asymptotic normality under censored data,etc,which haven’t been researched fore years.But these large sample properties play an important role in the theories of nonparametric regression estimation.In this paper, we mainly study three aspects as follows:（1）We obtain the strong consistency, mean and strong of the integrated absolute error consistency for partitioning estimate under α-mixing sample with identically distributed by means of the basic inequality for a - mixing sequence.（2）We obtain the strong consistency and convergence rate, mean and strong of the integrated absolute error consistency for modified partitioning estimate undera - mixing sample with identically distributed by means of Bernstein inequality for a - mixing sequence.（3）Based on censored sample, we prove the asymptotic normality for partitioning estimate and modified its estimate under suitable conditions by means of some properties of censored data and martingale theory.更多还原