相依样本下回归函数基于分割估计及其改良估计的统计推断理论
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摘要
设(X_1,Y_1),(X_2,Y_2),…(X_n,Y_n)为从取值于R~d×R~1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。
Let (X,Y) be a R~d×R~1 valued random vector and (X_1,Y_1), (X_2,Y_2) …(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 {(X_1,Y_1),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
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。
Let (X,Y) be a R~d×R~1 valued random vector and (X_1,Y_1), (X_2,Y_2) …(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 {(X_1,Y_1),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
a - 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.
引文
[1] Nadaraya, E. A., On estimating regression, Theory Probability Appl. 9 (1964), 141-142
[2] Watson, G. S., Smooth regression analysis, Sankhya, Serics. A, 26(1964), 359-372
[3] Devroye, L., On the almost everywhere convergence of nonparametric regression function estimates, Ann. Statist., 9(1981), 1310-1319
[4] Devroye. L. and Wagner. T. J., Distribution-free consistency results in nonparametric discrimination and regression function estimation, Ann. Statist., 8 (1980), 231-239
[5] Gyorfi. L., Hardle. W., Sarda. P., and Vieu. P., Nonparametric curve estimation from time series, Lecture Notes in Statistics, No. 60, Springer, Bedin. 1989
[6] Hardle. W., Janssen. P., and Serfling. R., Strong uniform consistency rates for estimator of conditional functions, Ann. Statist., 16(1988), 1428-1449
[7] Peligrad. M., Properties of uniform consistency of kernel estimators of density and regression functions under dependence assumptions, Stoch. Reports, 40 (1992), 147-168
[8] Greblicki. W., Krayzak. A., and Pawlak. M., Distribution-free pointwise consistency of kernel regression estimate, Ann. Statist., 12(4)(1984), 1570-1575
[9] 林正炎,关于回归函数核估计的渐近正态性,数学进展,16(1),(1984),97—102
[10] 胡舒合,非参数回归函数估计的渐近正态性,数学学报,45(3),(2002),433—442
[11] 成平,回归函数改良核估计的强相合性及收敛速度,系统科学与数学,3(4),(1983),304-315
[12] 胡舒合,回归函数改良核估计的相合性,系统科学与数学,13(2),(1993),141—151
[13] Qian, W. M. and Mammitizsch, V., Strong uniform convergence for the estimator of the regression function under φ-mixing conditions, Metdka, 52(1) (2000), 63-67
[14] 赵霞,回归函数改良核估计的渐近分布,应用概率统计,17(1),(2001),81-87
[15] Zheng Zu Kang, A Class of Estimators of the parameters in linear regression with censored data, Acta Mathematica Applicate Sinica, 3(3), (1987), 231-241
[16] 胡舒合,完全与截尾样本时回归函数的核估计,数学年刊,16A(3),(1995),269-274
[17] 胡舒合,分布自由的回归函数核估计的相合性,应用数学学报,20(3),(1997),448-455
[18] 赵林城,白志东,非参数回归函数最近邻核估计的强相合性,中国科学,5(5),(1984),387-393
[19] 洪圣岩,最近邻回归估计的渐近正态性,应用概率统计,7(2),(1991),187-191
[20] 成平,赵林城,回归函数之改良近邻估计的强相合性,科学通报,30(1),(1985),10-14
[21] 薛留根,相依样本时回归函数之两类改良估计的强相合性,数理统计与应用概率,8(1),(1993),82-93
[22] Paul Algoet and Laszlo Gyorfi, Strong universal pointwise consistency of some regression function estimates, J. of Multivariate Analysis, 71(1999), 125-144
[23] Ling Neng Xiang, Du Xue Qiao, Strong. Convergence of Partitioning Estimation for Nonparametric Regression Function Under Dependence Samples, Chinese Quarterly Journal of Mathematics, 20(1), (2005),, 28-33
[24] Ling Cheng Zhao,Guo Qing Diao,Strong.Convergence of Modified Partitioning Estimates of Nonparametric Regression Functions, Chinese Journal of Applied Probability and Statistics, 18(2), (2002), 192-196
[25] Ling Neng Xiang, Strong Convergence and Its Rate of Modified Partitioning Estimation for Nonparametric Regression Function under Dependence Samples, Northeast. Math. J, 20(3), (2004), 349-354
[26] 凌能祥,截尾数据时回归函数基于分割估计的相合性,应用概率统计,20(2),(2002),107-110
[27] 林正炎,陆传荣,混合相依变量的极限理论,北京:科学出版社,1997,5-6,8
[28] 杨善朝,α—混合序列和的强大数律及其应用,高校应用数学学报,11A(4),(1996),443-450
[29] 胡舒合,分布自由的回归函数近邻核估计的相合性,数学学报,38(4),(1995),559-567
[30] Boente.G. & Fraiman.R., Consistency of a nonparametric estimate of a density function for dependent variables, J. of Multivariate Analysis, 25(1988), 90-99
[31] Liptser S.R.,Shiryayev A.N., Theorem of Martingales, The Netherlands: Kluwer Academic Publishers, 1989
[32] 凌能祥,回归函数基于分割估计及其改良估计的渐近正态性,已投
[33] 姚梅,杜雪樵,α混合下非参数回归函数基于分割估计的强相合性,合肥工业大学学报(自然科学版),2006,5
[34] 姚梅,杜雪樵,α混合下非参数回归函数改良基于分割估计的强相合性及收敛速度,已投
[35] 陈希孺,王松桂,近代回归分析—原理方法及应用,安徽:安徽教育出版社,1987
[36] 陈希孺,方兆本等著,非参数统计,上海:上海科学技术出版社,1989
[37] 孙燕,柴根象,固定设计下回归函数的小波估计,数学物理学报,24A(5),(2004),597-606
[38] 许冰,强混合相依变量加权和的收敛性及应用,数学学报,45(5),(2002),1025-1034
[39] Roussas.G.G.and Tran.L.T., Asymptotic normality of the recursive kernel regression estimate under dependence conditions, Ann. Statist. 20(1), (1992), 98-120
[40] Rao.C.R., Linear Statistical Inference and Its Applications,2ed edition.,John Wiley & Sons,Inc.,1973
[2] Watson, G. S., Smooth regression analysis, Sankhya, Serics. A, 26(1964), 359-372
[3] Devroye, L., On the almost everywhere convergence of nonparametric regression function estimates, Ann. Statist., 9(1981), 1310-1319
[4] Devroye. L. and Wagner. T. J., Distribution-free consistency results in nonparametric discrimination and regression function estimation, Ann. Statist., 8 (1980), 231-239
[5] Gyorfi. L., Hardle. W., Sarda. P., and Vieu. P., Nonparametric curve estimation from time series, Lecture Notes in Statistics, No. 60, Springer, Bedin. 1989
[6] Hardle. W., Janssen. P., and Serfling. R., Strong uniform consistency rates for estimator of conditional functions, Ann. Statist., 16(1988), 1428-1449
[7] Peligrad. M., Properties of uniform consistency of kernel estimators of density and regression functions under dependence assumptions, Stoch. Reports, 40 (1992), 147-168
[8] Greblicki. W., Krayzak. A., and Pawlak. M., Distribution-free pointwise consistency of kernel regression estimate, Ann. Statist., 12(4)(1984), 1570-1575
[9] 林正炎,关于回归函数核估计的渐近正态性,数学进展,16(1),(1984),97—102
[10] 胡舒合,非参数回归函数估计的渐近正态性,数学学报,45(3),(2002),433—442
[11] 成平,回归函数改良核估计的强相合性及收敛速度,系统科学与数学,3(4),(1983),304-315
[12] 胡舒合,回归函数改良核估计的相合性,系统科学与数学,13(2),(1993),141—151
[13] Qian, W. M. and Mammitizsch, V., Strong uniform convergence for the estimator of the regression function under φ-mixing conditions, Metdka, 52(1) (2000), 63-67
[14] 赵霞,回归函数改良核估计的渐近分布,应用概率统计,17(1),(2001),81-87
[15] Zheng Zu Kang, A Class of Estimators of the parameters in linear regression with censored data, Acta Mathematica Applicate Sinica, 3(3), (1987), 231-241
[16] 胡舒合,完全与截尾样本时回归函数的核估计,数学年刊,16A(3),(1995),269-274
[17] 胡舒合,分布自由的回归函数核估计的相合性,应用数学学报,20(3),(1997),448-455
[18] 赵林城,白志东,非参数回归函数最近邻核估计的强相合性,中国科学,5(5),(1984),387-393
[19] 洪圣岩,最近邻回归估计的渐近正态性,应用概率统计,7(2),(1991),187-191
[20] 成平,赵林城,回归函数之改良近邻估计的强相合性,科学通报,30(1),(1985),10-14
[21] 薛留根,相依样本时回归函数之两类改良估计的强相合性,数理统计与应用概率,8(1),(1993),82-93
[22] Paul Algoet and Laszlo Gyorfi, Strong universal pointwise consistency of some regression function estimates, J. of Multivariate Analysis, 71(1999), 125-144
[23] Ling Neng Xiang, Du Xue Qiao, Strong. Convergence of Partitioning Estimation for Nonparametric Regression Function Under Dependence Samples, Chinese Quarterly Journal of Mathematics, 20(1), (2005),, 28-33
[24] Ling Cheng Zhao,Guo Qing Diao,Strong.Convergence of Modified Partitioning Estimates of Nonparametric Regression Functions, Chinese Journal of Applied Probability and Statistics, 18(2), (2002), 192-196
[25] Ling Neng Xiang, Strong Convergence and Its Rate of Modified Partitioning Estimation for Nonparametric Regression Function under Dependence Samples, Northeast. Math. J, 20(3), (2004), 349-354
[26] 凌能祥,截尾数据时回归函数基于分割估计的相合性,应用概率统计,20(2),(2002),107-110
[27] 林正炎,陆传荣,混合相依变量的极限理论,北京:科学出版社,1997,5-6,8
[28] 杨善朝,α—混合序列和的强大数律及其应用,高校应用数学学报,11A(4),(1996),443-450
[29] 胡舒合,分布自由的回归函数近邻核估计的相合性,数学学报,38(4),(1995),559-567
[30] Boente.G. & Fraiman.R., Consistency of a nonparametric estimate of a density function for dependent variables, J. of Multivariate Analysis, 25(1988), 90-99
[31] Liptser S.R.,Shiryayev A.N., Theorem of Martingales, The Netherlands: Kluwer Academic Publishers, 1989
[32] 凌能祥,回归函数基于分割估计及其改良估计的渐近正态性,已投
[33] 姚梅,杜雪樵,α混合下非参数回归函数基于分割估计的强相合性,合肥工业大学学报(自然科学版),2006,5
[34] 姚梅,杜雪樵,α混合下非参数回归函数改良基于分割估计的强相合性及收敛速度,已投
[35] 陈希孺,王松桂,近代回归分析—原理方法及应用,安徽:安徽教育出版社,1987
[36] 陈希孺,方兆本等著,非参数统计,上海:上海科学技术出版社,1989
[37] 孙燕,柴根象,固定设计下回归函数的小波估计,数学物理学报,24A(5),(2004),597-606
[38] 许冰,强混合相依变量加权和的收敛性及应用,数学学报,45(5),(2002),1025-1034
[39] Roussas.G.G.and Tran.L.T., Asymptotic normality of the recursive kernel regression estimate under dependence conditions, Ann. Statist. 20(1), (1992), 98-120
[40] Rao.C.R., Linear Statistical Inference and Its Applications,2ed edition.,John Wiley & Sons,Inc.,1973