混合误差过程回归模型小波估计Berry-Esseen界
|
摘要
本文在线性过程误差的假设下讨论了两个问题：一是非参数回归模型小波估计的Berry-Esseen界；二是半参数回归模型小波估计的Berry-Esseen界.
第一部分考虑非参数回归模型Uni=g(tni)+εi,i=1,…,n,其中9(·)是定义在区间[0,1]上的未知函数,{tni}是非随机设计点列,{εi}是随机误差.假设εi是由φ混合序列生成的线性误差,其中aj为常数且满足为φ混合序列.在上述假设下,我们建立了9(·)的小波估计的Berry-Esseen界.
第二部分考虑半参数回归模型Yi=xiβ+9(ti)+εi,i=1,…,n,此处β是一未知参数,g(·)是定义在区间[0,1]上的未知函数,{(xi,t)}是非随机设计点列,{Yi}是响应变量,{εi}是随机误差.假设εi是由强混合序列生成的线性误差,即其中aj为常数且满足：为强混合序列.在上述假设下,我们建立了未知参数β和g(·)的小波估计的Berry-Esseen界.
This dissertation addresses two issues under linear process errors:Berry-Esseen bounds for wavelet estimator in nonparametric regression model and semiparametric regression model.
In the first section, we consider the following nonparametric regression model:Yni=g(tni)+εi, i=1,…,n,were g(·) is an unknown function defined on the closed interval [0,1],{tni} are nonrandom design points,{εi} are random errors. Suppose that εi be an linear error generated by φ mixing, that is to say with∞, and {ei} are φ mixing.the Berry-Esseen type bound of wavelet estimator for the unknown function g(·) is established under the above assumption.
In the second section, we consider the following semiparametric regres-sion model:Yi=xiβ+g(ti)+εi,i=1,…,n,where β is an unknown param-eter of interest,{(xi, ti)} are nonrandom design points,{Yi} are the response variables, g(·) is an unknown function defined on the closed interval [0,1], and{εi} are random errors. Suppose that εi be an linear error generated by strong mixing, that is to say with and {ei} are strong mixing, the Berry-Esseen type bound of wavelet estimator for the unknown parameter β and the unknown function g(·) are established under the above assumption.
第一部分考虑非参数回归模型Uni=g(tni)+εi,i=1,…,n,其中9(·)是定义在区间[0,1]上的未知函数,{tni}是非随机设计点列,{εi}是随机误差.假设εi是由φ混合序列生成的线性误差,其中aj为常数且满足为φ混合序列.在上述假设下,我们建立了9(·)的小波估计的Berry-Esseen界.
第二部分考虑半参数回归模型Yi=xiβ+9(ti)+εi,i=1,…,n,此处β是一未知参数,g(·)是定义在区间[0,1]上的未知函数,{(xi,t)}是非随机设计点列,{Yi}是响应变量,{εi}是随机误差.假设εi是由强混合序列生成的线性误差,即其中aj为常数且满足：为强混合序列.在上述假设下,我们建立了未知参数β和g(·)的小波估计的Berry-Esseen界.
This dissertation addresses two issues under linear process errors:Berry-Esseen bounds for wavelet estimator in nonparametric regression model and semiparametric regression model.
In the first section, we consider the following nonparametric regression model:Yni=g(tni)+εi, i=1,…,n,were g(·) is an unknown function defined on the closed interval [0,1],{tni} are nonrandom design points,{εi} are random errors. Suppose that εi be an linear error generated by φ mixing, that is to say with∞, and {ei} are φ mixing.the Berry-Esseen type bound of wavelet estimator for the unknown function g(·) is established under the above assumption.
In the second section, we consider the following semiparametric regres-sion model:Yi=xiβ+g(ti)+εi,i=1,…,n,where β is an unknown param-eter of interest,{(xi, ti)} are nonrandom design points,{Yi} are the response variables, g(·) is an unknown function defined on the closed interval [0,1], and{εi} are random errors. Suppose that εi be an linear error generated by strong mixing, that is to say with and {ei} are strong mixing, the Berry-Esseen type bound of wavelet estimator for the unknown parameter β and the unknown function g(·) are established under the above assumption.
引文
[1]D. W. Andrews. Asymptotic normality of series estimators for nonparamet-ric and semiparametric regression models [J]. Econometrica:Journal of the Econometric Society, pages 307-345,1991.
[2]A. Antoniadis, G. Gregoire, and I. McKeague. Wavelet methods for curve estimation [J]. Journal of the American Statistical Association, 89(428):1340-1353,1994.
[3]B. Auestad and D. Tjiostheim. Identification of nonlinear time series:First or-der characterization and order determination [J]. Biometrika,77(4):669-687, 1990.
[4]G. Beylkin, R. Coifman, and V. Rokhlin. Fast wavelet transforms and nu-merical algorithms I. Communications on pure and applied mathematics, 44(2):141-183,1991.
[5]Y. V. Borovskikh and N. Weber. Berry-esseen type bounds for trimmed U-statistics[J]. Journal of Statistical Planning and Inference,138(4):1059-1071, 2008.
[6]A. W. Bowman and A. Azzalini. Applied Smoothing Techniques for Data Analysis:The Kernel Approach with S-Plus Illustrations[M]. Oxford Univer-sity Press,1997.
[7]R. J. Carroll and W. Haerdle. A note on second order effects in a semipara-metric context. Technical report, DTIC Document,1988.
[8]R. J. Carroll and D. Ruppert. Robust estimation in heteroscedastic linear models[J]. The annals of statistics,10(2):429-441,1982.
[9]G. Chen. Berry-esseen-type bounds for the kernel estimator of conditional distribution and conditional quantiles[J]. Journal of statistical planning and inference,60(2):311-330,1997.
[10]H. Chen and J. J. H. Shiau. Data-driven efficient estimators for a partially linear model[J]. The Annals of Statistics, pages 211-237,1994.
[11]C. Cheng. A berry-esseen-type theorem of quantile density estimators[J]. S-tatistics & probability letters,39(3):255-262,1998.
[12]R. Clark. Non-parametric estimation of a smooth regression function[J]. Journal of the Royal Statistical Society. Series B (Methodological), pages 107-113,1977.
[13]S. G. Donald and W. K. Newey. Series estimation of semilinear models[J]. Journal of Multivariate Analysis,50(1):30-40,1994.
[14]D. L. Donoho, I. M. Johnstone, G. Kerkyacharian, and D. Picard. Density estimation by wavelet thresholding. The Annals of Statistics, pages 508-539, 1996.
[15]R. F. Engle, C. W. Granger, J. Rice, and A. Weiss. Semiparametric esti-mates of the relation between weather and electricity sales [J]. Journal of the American Statistical Association,81 (394):310-320,1986.
[16]R. Eubank and P. Whitney. Convergence rates for estimation in certain par-tially linear models [J]. Journal of statistical planning and inference, 23(1):33-43,1989.
[17]R. L. Eubank and C. H. Spiegelman. Testing the goodness of fit of a linear model via nonparametric regression techniques [J]. Journal of the American Statistical Association,85(410):387-392,1990.
[18]J. Fan. Local polinomial modelling and its applications, volume 66[M]. CRC Press,1996.
[19]Y. Fan. Consistent nonparametric multiple regression for dependent hetero-geneous processes:the fixed design case[J]. Journal of multivariate analysis, 33(1):72-88,1990.
[20]W. A. Fuller and J. Rao. Estimation for a linear regression model with unknown diagonal covariance matrix[J]. The Annals of Statistics, pages 1149-1158,1978.
[21]J. Gao. Semiparametric regression smoothing of non-linear time series [J]. Scan-dinavian journal of statistics,25(3):521-539,1998.
[22]J. Gao and V. Anh. Semiparametric regression under long-range dependent errors[J]. Journal of statistical planning and inference,80(1):37-57,1999.
[23]J. Gao, H. Tong, and R. C. Wolff. Adaptive orthogonal series estimation in additive stochastic regression models[J]. Statistica Sinica,12(2):409-428, 2002.
[24]A. A. Georgiev. Consistent nonparametric multiple regression:the fixed de-sign case[J]. Journal of multivariate analysis,25(1):100--110,1988.
[25]Green P J, Silverman B W. Nonparametric regression and generalized linear models:a roughness penalty approach[M]. London:Chapman & Hall,1994.
[26]S. A. Hamilton and Y. K. Truong. Local linear estimation in partly linear models [J]. Journal of Multivariate Analysis,60(1):1-19,1997.
[27]W. Hardle. Applied nonparametric regression[M]. Cambridge Univ Press, 1990.
[28]W. Hardle. Smoothing techniques:with implementation in S[M]. Springer, 1991.
[29]W. Hardle and H. Liang. Partially linear models[M]. Springer,2007.
[30]R. Helmers and M. Huskova. The berry-esseen bound for u-statistics[M].1982.
[31]Hsu P L. The approximate distributions of the mean and variance of a sample of independent variables[J]. The Annals of Mathematical Statistics,1945, 16(1):1-29.
[32]S. Jaffard. Pointwise smoothness, two-microlocalization and wavelet coeffi-cients. Publicacions Matematiques,35(1):155-168,1991.
[33]S. Jaffard and P. Laurencot. Orthonormal wavelets, analysis of operators, and applications to numerical analysis. In Wavelets:a tutorial in theory and applications, pages 543-601. Academic Press Professional, Inc.,1993.
[34]Y. Li. Asymptotic normality of wavelet estimator for strong mixing errors[J]. Journal of the Korean Statistical Society,38(4):383-390,2009.
[35]Y. Li, C. Wei, and G. Xing. Berry-esseen bounds for wavelet estimator in a regression model with linear process errors[J]. Statistics & probability letters, 81(1):103-110,2011.
[36]Y. Li, C. Yin, and C. Wei. On the asymptotic normality for (?)-mixing depen-dent errors of wavelet regression function estimator[J]. Acta Math. Appl. Sin, 31(6):1046-1055,2008.
[37]H. Liang and Y. Qi. Asymptotic normality of wavelet estimator of regres-sion function under na assumptions [J]. BULLETIN-KOREAN MATHEMAT-ICAL SOCIETY,44(2):247,2007.
[38]H. Liang and G. Fan. Berry-esseen type boounds of estimators in a semipara-metric model with linear process errors [J]. Journal of Multivariate Analysis, 100(1):1-15,2009.
[39]H. Liang and B. Jing. Asymptotic properties for estimates of nonpara-met-ric regression models based on negatively associated sequences. Journal of multivariate analysis,95(2):227-245,2005.
[40]H. Liang and Y. Li. A berry-esseen type bound of regression estimator based on linear process errors[J]. Journal of the Korean Mathematical Society, 45(6):1753-1767,2008.
[41]L. Xue. Uniform convergence rates of the wavelet estimator of regression function under mixing error [J]. Acta Mathematiea Scientia,4:014,2002.
[42]T. K. Mak. Estimation of parameters in heteroscedastic linear models. Jour-nal of the Royal Statistical Society[J]. Series B (Methodological), pages 649-655,1992.
[43]H. G. Muller and U. Stadtmuller. Estimation of heteroscedasticity in regres-sion analysis[J]. The Annals of Statistics,15(2):610-625,1987.
[44]W. K. Newey. Convergence rates for series estimators[M].1993.
[45]W. K. Newey. Series estimation of regression functional [M], Econometric The-ory,10:1-28,1994.
[46]W. K. Newey. Convergence rates and asymptotic normality for series estima-tors[J]. Journal of Econometrics,79(1):147-168,1997.
[47]V. V. Petrov. Limit theorems of probability theory [M].1995.
[48]D. Pollard. Convergence of stochastic procresses[M]. David Pollard,1984.
[49]M. Priestley and M. Chao. Non-parametric function fitting. Journal of the Royal Statistical Society[J]. Series B (Methodological), pages 385-392,1972.
[50]G. G. Roussas, L. T. Tran, and D. Ioannides. Fixed design regression for time series:asymptotic normality [J]. Journal of Multivariate Analysis, 40(2):262-291,1992.
[51]M. B. Ruskai, G. Beylkin, and R. Coifman. Wavelets and their applications. Jones and Bartlett Books in Mathematics, Boston:Jones and Bartlett,1992, edited by Ruskai, Mary B.; Beylkin, Gregory; Coifman, Ronald,1,1992.
[52]A. Schick. Root-n consistent estimation in partly linear regression models[J]. Statistics & Probability Letters,28(4):353-358,1996.
[53]R. Schmalensee and T. M. Stoker. Household gasoline demand in the united states [J]. Econometrics,67(3):645-662,1999.
[54]P. Speckman. Kernel smoothing in partial linear models[J]. Journal of the Royal Statistical Society. Series B (Methodological), pages 413-436,1988.
[55]X. Sun, J. You, G. Chen, and X. Zhou. Convergence rates of estimators in partial linear regression models with ma(oo) error process[J]. Communications in Statistics-Theory and Methods,31(12):2251-2273,2002.
[56]L. Tran, G. Roussas, S. Yakowitz, and B. Truong Van. Fixed-design regression for linear time series[J]. The Annals of Statistics,24(3):975-991,1996.
[57]W. Van Zwet. A berry-esseen bound for symmetric statistics[J]. Zeitschrift fUr Wahrscheinlichkeitstheorie und verwandte Gebiete,66(3):425-440,1984.
[58]G. G. Walter. Wavelets and other orthogonal systems with applications[M]. CRC press,1994.
[59]M. Wand and M. Jones. Kernel smoothing, vol.60 of[M]. Monographs on statistics and applied probability,1995.
[60]M. V. Wickerhauser. Adapted wavelet analysis from theory to software.1994.
[61]C. M. Wong and R. Kohn. A bayesian approach to estimating and forecast-ing additive nonparametric autoregressive models [J]. Journal of Time Series Analysis,17(2):203-220,1996.
[62]S. Yang. Almost sure convergence of weighted sums of mixing sequences[J]. J. Syst. Sci. Math. Sci,15(3):254-265,1995.
[63]S. Yang. Uniformly asymptotic normality of the regression weighted esti-mator for negatively associated samples [J]. Statistics & probability letters, 62(2):101-110,2003.
[64]S. Yang. Maximal moment inequality for partial sums of strong mix-in-g sequences and application[J]. Acta Mathematica Sinica, English Series, 23(6):1013-1024,2007.
[65]J. You, M. Chen, and G. Chen. Asymptotic normality of some estimators in a fixed-design semiparametric regression model with linear time series errors [J]. J Syst Sci Complex,17(4):511-522,2004.
[66]Z. Lin, Z. Lin, and C. Lu. Limit theory for mixing dependent random vari-ables[M]. Springer,1996.
[67]柴根象.半参数回归模型[M].安徽教育出版社,1995.
[68]柴根象,刘元金.固定设计下混合序列回归函数的小波估计[J].数学研究与评论,2001,21(4)：555-560.
[69]柴根象,徐克军.半参数回归的线性小波光滑[J].应用概率统计,1999,15：97-106.
[70]陈希孺.U-统计量和Von-Mises统计量的极限性质[J].中国科学,1986(3)：522.
[71]陈希孺.线性模型参数的估计理论[M].科学出版社,2010.
[72]陈希孺.数理统计引论[M].科学出版社,2007.
[73]陈希孺,方兆本,李国英,陶波.非参数统计[M].中国科技大学出版社,2012.
[74]黄四民,梁华.用半参数部分线性模型分析居民消费结构[J].数量经济技术经济研究,1994,10：33-39.
[75]姜玉英,刘强,吴可法.半参数回归模型小波估计的相合性[J].福州大学学报(自然科学版),2006,34：798-803.
[76]金剑,蒋萍.生产率增长测算的半参数估计方法：理论综述和相关探讨[J].数量经济技术经济研究.2006(09).
[77]孔燕,柴根象.固定设计下混合序列回归函数的小波估计[J].数学物理学报,2004,24A(5)：597-606.
[78]李永明,韦程东.强混合误差回归函数小波估计的Berry-Esseen界[J].数学物理学报,2009(5).
[79]李永明,尹长明,韦程东.妒混合误差下回归函数小波估计的渐近正态性[J].应用数学学报,2008,31(6)：1046-1055.
[80]梁华,熊健.再论半参数部分线性模型在居民消费结构分析中的应用[J].数理统计与管理,1995,6：9-17.
[81]林正炎. Student U-统计量的Berry-Esseen界[J].中国科学(A辑),2000,30(4)：307-316.
[82]林正炎,陆传荣,苏中根.概率极限理论基础[M].高等教育出版社,1999.
[83]林正炎,陆传荣.强极限理论[M].科学出版社,1992.
[84]陆传荣,林正炎.概率论极限理论引论[M].高等教育出版社,1989.
[85]陆传荣,林正炎.混合相依变量的极限理论[M].科学出版社,1997.
[86]钱伟民,柴根象.半参数回归模型小波估计的强逼近[J].中国科学(A),1999,3：233-241.
[87]钱伟民,柴根象,蒋凤瑛.半参数回归模型的误差方差的小波估计[J].数学年刊(A),2000,3：341-350.
[88]施锡铨.广义U统计量正态逼近的速度[J].数学杂志,1986,6(3)：237-247.
[89]孙六全,朱力行.随机删失数据下核密度估计的Berry-Esseen界[J].数学学报,1999,42(4).
[90]唐湘晋.关于U-统计量的三角格式的Berry-Esseen界限[J].武汉水运工程学院学报,1988,12(3).
[91]田阿芳,王文胜.随机样本中最大点个数的Berry-Esseen界[J].杭州师范学院学报(自然科学版),2004,3(2)：95-97.
[92]田萍.纵向数据半参数回归模型的估计理论[M].郑州大学出版社,2008.
[93]王仁宏,崔锦泰.关于一个二元B样条基[J].中国科学(A辑数学物理学天文学技术科学),1984,9：001.
[94]汪书润,凌能祥.半参数回归模型小波估计的强相合性[J].合肥工业大学学报,2009,32(1)：136-138.
[95]王松桂.线性统计模型线性回归与方差分析[M].高等教育出版社,1999.
[96]王松桂,史建红,尹索菊,吴密霞.线性模型引论[M].科学出版社,2004.
[97]王一兵.半参数回归模型在商品房价格指数中的应用研究[J].统计研究,2005,4：25-30.
[98]吴群英.混合序列的概率极限理论[M].科学出版社,2006.
[99]许冰,陈娟.浙江省城镇居民消费与收入的半参数线性回归模型[J].浙江统计,2004(06).
[100]薛留根.混合误差下回归函数小波估计的一致收敛速度[J].数学物理学报,2002,22A(4)：528-535.
[101]薛留根.半参数回归模型中小波估计的随机加权逼近速度[J].应用数学学报,2003,26：11-26.
[102]薛留根.半参数回归模型中误差方差估计的Berry-Esseen界[J].数学学报,2005,48(1)：157-170.
[103]严士健,王隽骧,刘秀芳.概率论基础[M].科学出版社,1982.
[104]杨善朝.强混合序列的矩不等式及其应用[J].数学研究与评论,2000,20(3)：349-359.
[105]杨善朝,李永明.强混合样本下回归加权估计的一致渐近正态性[J].数学学报,2006,49A(5)1163-1170.
[106]赵林城,陈希孺.有限总体U-统计量的Berry-Esseen界限[J].中国科学,1985.
[2]A. Antoniadis, G. Gregoire, and I. McKeague. Wavelet methods for curve estimation [J]. Journal of the American Statistical Association, 89(428):1340-1353,1994.
[3]B. Auestad and D. Tjiostheim. Identification of nonlinear time series:First or-der characterization and order determination [J]. Biometrika,77(4):669-687, 1990.
[4]G. Beylkin, R. Coifman, and V. Rokhlin. Fast wavelet transforms and nu-merical algorithms I. Communications on pure and applied mathematics, 44(2):141-183,1991.
[5]Y. V. Borovskikh and N. Weber. Berry-esseen type bounds for trimmed U-statistics[J]. Journal of Statistical Planning and Inference,138(4):1059-1071, 2008.
[6]A. W. Bowman and A. Azzalini. Applied Smoothing Techniques for Data Analysis:The Kernel Approach with S-Plus Illustrations[M]. Oxford Univer-sity Press,1997.
[7]R. J. Carroll and W. Haerdle. A note on second order effects in a semipara-metric context. Technical report, DTIC Document,1988.
[8]R. J. Carroll and D. Ruppert. Robust estimation in heteroscedastic linear models[J]. The annals of statistics,10(2):429-441,1982.
[9]G. Chen. Berry-esseen-type bounds for the kernel estimator of conditional distribution and conditional quantiles[J]. Journal of statistical planning and inference,60(2):311-330,1997.
[10]H. Chen and J. J. H. Shiau. Data-driven efficient estimators for a partially linear model[J]. The Annals of Statistics, pages 211-237,1994.
[11]C. Cheng. A berry-esseen-type theorem of quantile density estimators[J]. S-tatistics & probability letters,39(3):255-262,1998.
[12]R. Clark. Non-parametric estimation of a smooth regression function[J]. Journal of the Royal Statistical Society. Series B (Methodological), pages 107-113,1977.
[13]S. G. Donald and W. K. Newey. Series estimation of semilinear models[J]. Journal of Multivariate Analysis,50(1):30-40,1994.
[14]D. L. Donoho, I. M. Johnstone, G. Kerkyacharian, and D. Picard. Density estimation by wavelet thresholding. The Annals of Statistics, pages 508-539, 1996.
[15]R. F. Engle, C. W. Granger, J. Rice, and A. Weiss. Semiparametric esti-mates of the relation between weather and electricity sales [J]. Journal of the American Statistical Association,81 (394):310-320,1986.
[16]R. Eubank and P. Whitney. Convergence rates for estimation in certain par-tially linear models [J]. Journal of statistical planning and inference, 23(1):33-43,1989.
[17]R. L. Eubank and C. H. Spiegelman. Testing the goodness of fit of a linear model via nonparametric regression techniques [J]. Journal of the American Statistical Association,85(410):387-392,1990.
[18]J. Fan. Local polinomial modelling and its applications, volume 66[M]. CRC Press,1996.
[19]Y. Fan. Consistent nonparametric multiple regression for dependent hetero-geneous processes:the fixed design case[J]. Journal of multivariate analysis, 33(1):72-88,1990.
[20]W. A. Fuller and J. Rao. Estimation for a linear regression model with unknown diagonal covariance matrix[J]. The Annals of Statistics, pages 1149-1158,1978.
[21]J. Gao. Semiparametric regression smoothing of non-linear time series [J]. Scan-dinavian journal of statistics,25(3):521-539,1998.
[22]J. Gao and V. Anh. Semiparametric regression under long-range dependent errors[J]. Journal of statistical planning and inference,80(1):37-57,1999.
[23]J. Gao, H. Tong, and R. C. Wolff. Adaptive orthogonal series estimation in additive stochastic regression models[J]. Statistica Sinica,12(2):409-428, 2002.
[24]A. A. Georgiev. Consistent nonparametric multiple regression:the fixed de-sign case[J]. Journal of multivariate analysis,25(1):100--110,1988.
[25]Green P J, Silverman B W. Nonparametric regression and generalized linear models:a roughness penalty approach[M]. London:Chapman & Hall,1994.
[26]S. A. Hamilton and Y. K. Truong. Local linear estimation in partly linear models [J]. Journal of Multivariate Analysis,60(1):1-19,1997.
[27]W. Hardle. Applied nonparametric regression[M]. Cambridge Univ Press, 1990.
[28]W. Hardle. Smoothing techniques:with implementation in S[M]. Springer, 1991.
[29]W. Hardle and H. Liang. Partially linear models[M]. Springer,2007.
[30]R. Helmers and M. Huskova. The berry-esseen bound for u-statistics[M].1982.
[31]Hsu P L. The approximate distributions of the mean and variance of a sample of independent variables[J]. The Annals of Mathematical Statistics,1945, 16(1):1-29.
[32]S. Jaffard. Pointwise smoothness, two-microlocalization and wavelet coeffi-cients. Publicacions Matematiques,35(1):155-168,1991.
[33]S. Jaffard and P. Laurencot. Orthonormal wavelets, analysis of operators, and applications to numerical analysis. In Wavelets:a tutorial in theory and applications, pages 543-601. Academic Press Professional, Inc.,1993.
[34]Y. Li. Asymptotic normality of wavelet estimator for strong mixing errors[J]. Journal of the Korean Statistical Society,38(4):383-390,2009.
[35]Y. Li, C. Wei, and G. Xing. Berry-esseen bounds for wavelet estimator in a regression model with linear process errors[J]. Statistics & probability letters, 81(1):103-110,2011.
[36]Y. Li, C. Yin, and C. Wei. On the asymptotic normality for (?)-mixing depen-dent errors of wavelet regression function estimator[J]. Acta Math. Appl. Sin, 31(6):1046-1055,2008.
[37]H. Liang and Y. Qi. Asymptotic normality of wavelet estimator of regres-sion function under na assumptions [J]. BULLETIN-KOREAN MATHEMAT-ICAL SOCIETY,44(2):247,2007.
[38]H. Liang and G. Fan. Berry-esseen type boounds of estimators in a semipara-metric model with linear process errors [J]. Journal of Multivariate Analysis, 100(1):1-15,2009.
[39]H. Liang and B. Jing. Asymptotic properties for estimates of nonpara-met-ric regression models based on negatively associated sequences. Journal of multivariate analysis,95(2):227-245,2005.
[40]H. Liang and Y. Li. A berry-esseen type bound of regression estimator based on linear process errors[J]. Journal of the Korean Mathematical Society, 45(6):1753-1767,2008.
[41]L. Xue. Uniform convergence rates of the wavelet estimator of regression function under mixing error [J]. Acta Mathematiea Scientia,4:014,2002.
[42]T. K. Mak. Estimation of parameters in heteroscedastic linear models. Jour-nal of the Royal Statistical Society[J]. Series B (Methodological), pages 649-655,1992.
[43]H. G. Muller and U. Stadtmuller. Estimation of heteroscedasticity in regres-sion analysis[J]. The Annals of Statistics,15(2):610-625,1987.
[44]W. K. Newey. Convergence rates for series estimators[M].1993.
[45]W. K. Newey. Series estimation of regression functional [M], Econometric The-ory,10:1-28,1994.
[46]W. K. Newey. Convergence rates and asymptotic normality for series estima-tors[J]. Journal of Econometrics,79(1):147-168,1997.
[47]V. V. Petrov. Limit theorems of probability theory [M].1995.
[48]D. Pollard. Convergence of stochastic procresses[M]. David Pollard,1984.
[49]M. Priestley and M. Chao. Non-parametric function fitting. Journal of the Royal Statistical Society[J]. Series B (Methodological), pages 385-392,1972.
[50]G. G. Roussas, L. T. Tran, and D. Ioannides. Fixed design regression for time series:asymptotic normality [J]. Journal of Multivariate Analysis, 40(2):262-291,1992.
[51]M. B. Ruskai, G. Beylkin, and R. Coifman. Wavelets and their applications. Jones and Bartlett Books in Mathematics, Boston:Jones and Bartlett,1992, edited by Ruskai, Mary B.; Beylkin, Gregory; Coifman, Ronald,1,1992.
[52]A. Schick. Root-n consistent estimation in partly linear regression models[J]. Statistics & Probability Letters,28(4):353-358,1996.
[53]R. Schmalensee and T. M. Stoker. Household gasoline demand in the united states [J]. Econometrics,67(3):645-662,1999.
[54]P. Speckman. Kernel smoothing in partial linear models[J]. Journal of the Royal Statistical Society. Series B (Methodological), pages 413-436,1988.
[55]X. Sun, J. You, G. Chen, and X. Zhou. Convergence rates of estimators in partial linear regression models with ma(oo) error process[J]. Communications in Statistics-Theory and Methods,31(12):2251-2273,2002.
[56]L. Tran, G. Roussas, S. Yakowitz, and B. Truong Van. Fixed-design regression for linear time series[J]. The Annals of Statistics,24(3):975-991,1996.
[57]W. Van Zwet. A berry-esseen bound for symmetric statistics[J]. Zeitschrift fUr Wahrscheinlichkeitstheorie und verwandte Gebiete,66(3):425-440,1984.
[58]G. G. Walter. Wavelets and other orthogonal systems with applications[M]. CRC press,1994.
[59]M. Wand and M. Jones. Kernel smoothing, vol.60 of[M]. Monographs on statistics and applied probability,1995.
[60]M. V. Wickerhauser. Adapted wavelet analysis from theory to software.1994.
[61]C. M. Wong and R. Kohn. A bayesian approach to estimating and forecast-ing additive nonparametric autoregressive models [J]. Journal of Time Series Analysis,17(2):203-220,1996.
[62]S. Yang. Almost sure convergence of weighted sums of mixing sequences[J]. J. Syst. Sci. Math. Sci,15(3):254-265,1995.
[63]S. Yang. Uniformly asymptotic normality of the regression weighted esti-mator for negatively associated samples [J]. Statistics & probability letters, 62(2):101-110,2003.
[64]S. Yang. Maximal moment inequality for partial sums of strong mix-in-g sequences and application[J]. Acta Mathematica Sinica, English Series, 23(6):1013-1024,2007.
[65]J. You, M. Chen, and G. Chen. Asymptotic normality of some estimators in a fixed-design semiparametric regression model with linear time series errors [J]. J Syst Sci Complex,17(4):511-522,2004.
[66]Z. Lin, Z. Lin, and C. Lu. Limit theory for mixing dependent random vari-ables[M]. Springer,1996.
[67]柴根象.半参数回归模型[M].安徽教育出版社,1995.
[68]柴根象,刘元金.固定设计下混合序列回归函数的小波估计[J].数学研究与评论,2001,21(4)：555-560.
[69]柴根象,徐克军.半参数回归的线性小波光滑[J].应用概率统计,1999,15：97-106.
[70]陈希孺.U-统计量和Von-Mises统计量的极限性质[J].中国科学,1986(3)：522.
[71]陈希孺.线性模型参数的估计理论[M].科学出版社,2010.
[72]陈希孺.数理统计引论[M].科学出版社,2007.
[73]陈希孺,方兆本,李国英,陶波.非参数统计[M].中国科技大学出版社,2012.
[74]黄四民,梁华.用半参数部分线性模型分析居民消费结构[J].数量经济技术经济研究,1994,10：33-39.
[75]姜玉英,刘强,吴可法.半参数回归模型小波估计的相合性[J].福州大学学报(自然科学版),2006,34：798-803.
[76]金剑,蒋萍.生产率增长测算的半参数估计方法：理论综述和相关探讨[J].数量经济技术经济研究.2006(09).
[77]孔燕,柴根象.固定设计下混合序列回归函数的小波估计[J].数学物理学报,2004,24A(5)：597-606.
[78]李永明,韦程东.强混合误差回归函数小波估计的Berry-Esseen界[J].数学物理学报,2009(5).
[79]李永明,尹长明,韦程东.妒混合误差下回归函数小波估计的渐近正态性[J].应用数学学报,2008,31(6)：1046-1055.
[80]梁华,熊健.再论半参数部分线性模型在居民消费结构分析中的应用[J].数理统计与管理,1995,6：9-17.
[81]林正炎. Student U-统计量的Berry-Esseen界[J].中国科学(A辑),2000,30(4)：307-316.
[82]林正炎,陆传荣,苏中根.概率极限理论基础[M].高等教育出版社,1999.
[83]林正炎,陆传荣.强极限理论[M].科学出版社,1992.
[84]陆传荣,林正炎.概率论极限理论引论[M].高等教育出版社,1989.
[85]陆传荣,林正炎.混合相依变量的极限理论[M].科学出版社,1997.
[86]钱伟民,柴根象.半参数回归模型小波估计的强逼近[J].中国科学(A),1999,3：233-241.
[87]钱伟民,柴根象,蒋凤瑛.半参数回归模型的误差方差的小波估计[J].数学年刊(A),2000,3：341-350.
[88]施锡铨.广义U统计量正态逼近的速度[J].数学杂志,1986,6(3)：237-247.
[89]孙六全,朱力行.随机删失数据下核密度估计的Berry-Esseen界[J].数学学报,1999,42(4).
[90]唐湘晋.关于U-统计量的三角格式的Berry-Esseen界限[J].武汉水运工程学院学报,1988,12(3).
[91]田阿芳,王文胜.随机样本中最大点个数的Berry-Esseen界[J].杭州师范学院学报(自然科学版),2004,3(2)：95-97.
[92]田萍.纵向数据半参数回归模型的估计理论[M].郑州大学出版社,2008.
[93]王仁宏,崔锦泰.关于一个二元B样条基[J].中国科学(A辑数学物理学天文学技术科学),1984,9：001.
[94]汪书润,凌能祥.半参数回归模型小波估计的强相合性[J].合肥工业大学学报,2009,32(1)：136-138.
[95]王松桂.线性统计模型线性回归与方差分析[M].高等教育出版社,1999.
[96]王松桂,史建红,尹索菊,吴密霞.线性模型引论[M].科学出版社,2004.
[97]王一兵.半参数回归模型在商品房价格指数中的应用研究[J].统计研究,2005,4：25-30.
[98]吴群英.混合序列的概率极限理论[M].科学出版社,2006.
[99]许冰,陈娟.浙江省城镇居民消费与收入的半参数线性回归模型[J].浙江统计,2004(06).
[100]薛留根.混合误差下回归函数小波估计的一致收敛速度[J].数学物理学报,2002,22A(4)：528-535.
[101]薛留根.半参数回归模型中小波估计的随机加权逼近速度[J].应用数学学报,2003,26：11-26.
[102]薛留根.半参数回归模型中误差方差估计的Berry-Esseen界[J].数学学报,2005,48(1)：157-170.
[103]严士健,王隽骧,刘秀芳.概率论基础[M].科学出版社,1982.
[104]杨善朝.强混合序列的矩不等式及其应用[J].数学研究与评论,2000,20(3)：349-359.
[105]杨善朝,李永明.强混合样本下回归加权估计的一致渐近正态性[J].数学学报,2006,49A(5)1163-1170.
[106]赵林城,陈希孺.有限总体U-统计量的Berry-Esseen界限[J].中国科学,1985.