相依序列下Gass-Muller回归权估计的大样本性质
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摘要
非参数回归估计是研究回归模型的一种有用工具,在金融经济方面有重要应用,如在金融资产价格和收益率波动性等方面有重要的的研究应用.在非参数回归估计中,通常采用权函数回归估计.自Sotne(1977)提出非参数回归估计的权函数估计方法后,其方法引起了广泛的重视.对于固定设计回归模型Yi=g(xi)+εi,1≤i≤n,Gass and Muller[1](1979)引入了权函数从而称为Gasser-Muller型权函数回归估计,其中K(·)是Borel可测函数,0 自从Gasser and Miiller提出Gasser-Muller回归权估计后,相继有一些学者对Gasser-Muller回归权估计进行了研究.Roussas, G.G., Tran, L.T.and Ioannides,D.A[3](1992)在a混合序列下讨论了该估计的渐近正态性,但没有给出收敛速度.Jianqing Fan[4](1992)通过两个模型改变有限样本容量进行模拟,比较了Gasser-Muller权估计,Nadaraya-Watson回归权估计和局部线性回归估计三种估计的均方误差及线性光滑性.Xiaoling Dou and Shongo Shirahata[5](2009)采用Gasser-Muller回归权估计用于一些评价基准和进行设计程序编程的模拟,并得出Gasser-Muller回归权估计的模拟结果优于局部多项式估计.
然而Gasser-Muller回归权估计的理论研究却比较少,因此,研究该估计不仅可以完善非参数回归的理论,还有着重要的现实意义.本文在在ρ混合序列和NOD序列情形下研究Gasser-Muller回归权估计的大样本性质,研究的主要内容和结果如下：
首先,讨论在样本为ρ混合序列和NOD序列情形下Gasser-Muller回归权估计的强相合性,推广了文献[13]中的推论,并减弱了文献[7]结论的条件.
其次,讨论在样本为ρ混合序列情形下Gasser-Muller回归权估计的一致渐近正态性,并给出一致渐近正态性的收敛速度,其收敛速度约为n-1/6.
再次,对一些ρ混合序列和NOD序列的Gasser-Muller回归权估计进行数值模拟.由数值模拟结果发现,当样本容量越多时,估计的误差越小,精度越高.
最后,选取我国股市的上证180和上证金融进行实证研究.
The nonparametric regression estimation is a useful tool in researching regression model. It is applied widely in financial economics, such as financial asset price and volatility of return rate. In nonparametric regression estimation, the weighted estimator of regression function is always employed. Since Sotne (1977) proposed the weight function of estimation method which is belonged to nonparametric regression estimates, this method is caused a wide attention. For fixed design regression model Yi=g(xi)+εi,1≤i≤n, Gasser and Muller[1](1979) introduced the weight function is called as Gasser-Miiller type weight function regression estimation, where K(-) is Borel mea-surable function,0 However the theory of the Gasser-Muller regression weighted estimation is researched sel-dom. Therefore, to research its theory not only can improve the nonparametric regression theory but also has an important practical significance to research this estimates.
In this paper, the author studies the large sample properties of the Gasser-Muller regression weighted estimate for the p mixed sequence and NOD sequence case. And the main research contents and the results are as follows:
First of all, discuss the strong consistency of the Gasser-Muller regression right estimate of the strong consistency with the help of in the sample for p mixed sequence and NOD sequence case. extend the corollaries of theorems in literature[13] and weaken the conditions of the theorem in literature[7].
Second, discuss the uniformly asymptotic normality of the Gasser-Muller regression weighted estimate for p mixed sequence case. And then provide the uniformly asymptotic normality con-vergence speed. Its convergence rate is about n-1/6.
Third, do some numerical simulation for the p mixed sequence and NOD sequence. From the numerical simulation results, the author finds that when the sample size is more, the estimated error is smaller and the accuracy is higher.
Finally, the author selects Shanghai Stock180index of Chinese stock market and the Shanghai Stock financial index to do empirical research.
然而Gasser-Muller回归权估计的理论研究却比较少,因此,研究该估计不仅可以完善非参数回归的理论,还有着重要的现实意义.本文在在ρ混合序列和NOD序列情形下研究Gasser-Muller回归权估计的大样本性质,研究的主要内容和结果如下：
首先,讨论在样本为ρ混合序列和NOD序列情形下Gasser-Muller回归权估计的强相合性,推广了文献[13]中的推论,并减弱了文献[7]结论的条件.
其次,讨论在样本为ρ混合序列情形下Gasser-Muller回归权估计的一致渐近正态性,并给出一致渐近正态性的收敛速度,其收敛速度约为n-1/6.
再次,对一些ρ混合序列和NOD序列的Gasser-Muller回归权估计进行数值模拟.由数值模拟结果发现,当样本容量越多时,估计的误差越小,精度越高.
最后,选取我国股市的上证180和上证金融进行实证研究.
The nonparametric regression estimation is a useful tool in researching regression model. It is applied widely in financial economics, such as financial asset price and volatility of return rate. In nonparametric regression estimation, the weighted estimator of regression function is always employed. Since Sotne (1977) proposed the weight function of estimation method which is belonged to nonparametric regression estimates, this method is caused a wide attention. For fixed design regression model Yi=g(xi)+εi,1≤i≤n, Gasser and Muller[1](1979) introduced the weight function is called as Gasser-Miiller type weight function regression estimation, where K(-) is Borel mea-surable function,0
In this paper, the author studies the large sample properties of the Gasser-Muller regression weighted estimate for the p mixed sequence and NOD sequence case. And the main research contents and the results are as follows:
First of all, discuss the strong consistency of the Gasser-Muller regression right estimate of the strong consistency with the help of in the sample for p mixed sequence and NOD sequence case. extend the corollaries of theorems in literature[13] and weaken the conditions of the theorem in literature[7].
Second, discuss the uniformly asymptotic normality of the Gasser-Muller regression weighted estimate for p mixed sequence case. And then provide the uniformly asymptotic normality con-vergence speed. Its convergence rate is about n-1/6.
Third, do some numerical simulation for the p mixed sequence and NOD sequence. From the numerical simulation results, the author finds that when the sample size is more, the estimated error is smaller and the accuracy is higher.
Finally, the author selects Shanghai Stock180index of Chinese stock market and the Shanghai Stock financial index to do empirical research.
引文
[1]Gass T., Muller H. G.. Kernels estimation of regression function[J]. Lecture Notes in Math-ematics,1979,757:23-68.
[2]Priestley, M. B., Chao, M. T.. Nonpar ametric function fitting[J]. J R Statist. Soc. (Series B), 1972,34:385-392.
[3]Roussas, G. G., Tran, L.T., Ioannides, D.A.. Fixed design regression for time series:Asymp-totic normality[J]. Journal of Multivariate Analysis,1992,40:262-291.
[4]Jianqing Fan. Design-adaptive nonparametric regression[J]. Journal of the American Statis-tical Association,1992,87:998-1004.
[5]Xiaoling Dou, Shongo Shirahata. Comparisons of b-spline procedures with kernel proce-dures in estimating regression functions and their derivatives[J]. J. Jpn. Conp. Statist,2009, 22:57-77.
[6]Benedetti, J. K.. On the nonparametric estimation of regression function[J]. J. R. Statist. Soc. B,1977,39:248-253.
[7]Schuster, E., Yakowitz, S.. Contribution to the theory of nonparametric regression, with application to aystem identification[J]. Ann. Statist,1979,7(1):139-149.
[8]秦永松.非参数回归函数估计的一个结果[J].工程数学学报,1989,6(3)：120-123.
[9]秦永松.相依误差下非参数回归函数估计的强相合性[J].广西师范大学学报,1992,10(2),24-27.
[10]于德明,王志江.φ混合误差下回归函数加权核估计的强相合性[J].中国计量学院学报,1994,7(1),1-4.
[11]杨善朝.φ混合误差下非参数回归函数加权核估计的相合性[J].高校应用数学学报,1995,10(2)：231-238.
[12]杨善朝.混合序列矩不等式和非参数估计[J].数学学报,1997,40(2)：271-279.
[13]杨善朝,王岳宝.NA样本回归函数估计的强相合性[J],应用数学学报,1999,22(4)：522-530.
[14]于德明,王勤,旺珍娥.a混合误差下回归函数加权核估计的强相合性[J].浙江工业大学学报,2006,34(4)：470-472.
[15]潘丽静,郭鹏江,白美利.ρ合误差下回归权函数估计的强相合性[J].西北大学学报,2010,40(5)：756-759.
[16) Kolmogorov A. N., Rozanov U.A.. On the strong mixing conditions for stationary gaussion process[J]. Theor. Probab. Appl.1960,5:204-208.
[17]Rosenblatt, M.. Markov processes, structure and asymptotic behavior[M]. New York: Springer-Verlag,1971,216-217.
[18]Bradley, R. C. Equivalent mixing conditions for markov chains[J]. Statist. Probab. Letters, 1999,41(1999):97-99.
[19]Joag-Dev K., Proschan F.. Negative association of random variables with application[J]. Ann Statist,1983,11:286-295.
[20]邵启满.关于ρ混合序列的完全收敛性[J].数学学报,1989,32(3)：372-393.
[21 ]杨文志.NOD序列的指数不等式及其应用[D].安徽：安徽大学,2010.
[22]罗中德.ρ混合序列下估计的渐近性质[D].广西：广西师范大学,2010.
[23]Yang S. C. Uniformly asymptotic normality of regression weighted estimator for negatively associated sample[J]. Statist. Probab. Lett,2003,62,101-110.
[24]Kolmogorov A. N., Rozanov U. A.. On the strong mixing conditions for stationary gaussion process[J]. Theor. Probab. Appl,1960,5,204-208.
[25]Peter J., Brockwell, Richard A. Davis.. Time series:therory and meth-ods[M]. Heidelberg: Springer Verlag,1998.
[26]王学军.弱鞅和三类相依序列的概率不等式及极限定理[D].安徽：安徽大学,2010.
[27]杨善朝,李永明.强混合样本下回归加权估计的一致渐进正态性[J].数学学报,2006(5)：1164-1170
[28]李军,杨善朝.NA样本下固定设计回归估计的渐进正态性[J].工程数学学报,2004,21(1)：97-102.
[29]Yang S.C.. Moment bounds for strong mixing sequences and their application [J]. Mathemat-ical Rasearch and Exposition,2000,20(3):349-359.
[30]George G. Roussas.. Fixed design regression for time series:asymptotic normality[J]. Jour-nal of Multivariate Analysis,1992,4:262-291.
[31]胡舒合.固定设计下的非参数多元回归[J].科学通报,1989,16：1275-1276.
[32]苏淳,赵林城,王岳宝.NA序列的矩不等式与弱收敛[J].中国科学(A),1996,26(12)：1091-1099.
[33]林正炎,陆传荣,苏中根.概率极限理论基础[M].北京：高等教育出版社,1999.
[34]王成名,余鑫晖.应用概率统计[M].广西师范大学出版社,2003.
[2]Priestley, M. B., Chao, M. T.. Nonpar ametric function fitting[J]. J R Statist. Soc. (Series B), 1972,34:385-392.
[3]Roussas, G. G., Tran, L.T., Ioannides, D.A.. Fixed design regression for time series:Asymp-totic normality[J]. Journal of Multivariate Analysis,1992,40:262-291.
[4]Jianqing Fan. Design-adaptive nonparametric regression[J]. Journal of the American Statis-tical Association,1992,87:998-1004.
[5]Xiaoling Dou, Shongo Shirahata. Comparisons of b-spline procedures with kernel proce-dures in estimating regression functions and their derivatives[J]. J. Jpn. Conp. Statist,2009, 22:57-77.
[6]Benedetti, J. K.. On the nonparametric estimation of regression function[J]. J. R. Statist. Soc. B,1977,39:248-253.
[7]Schuster, E., Yakowitz, S.. Contribution to the theory of nonparametric regression, with application to aystem identification[J]. Ann. Statist,1979,7(1):139-149.
[8]秦永松.非参数回归函数估计的一个结果[J].工程数学学报,1989,6(3)：120-123.
[9]秦永松.相依误差下非参数回归函数估计的强相合性[J].广西师范大学学报,1992,10(2),24-27.
[10]于德明,王志江.φ混合误差下回归函数加权核估计的强相合性[J].中国计量学院学报,1994,7(1),1-4.
[11]杨善朝.φ混合误差下非参数回归函数加权核估计的相合性[J].高校应用数学学报,1995,10(2)：231-238.
[12]杨善朝.混合序列矩不等式和非参数估计[J].数学学报,1997,40(2)：271-279.
[13]杨善朝,王岳宝.NA样本回归函数估计的强相合性[J],应用数学学报,1999,22(4)：522-530.
[14]于德明,王勤,旺珍娥.a混合误差下回归函数加权核估计的强相合性[J].浙江工业大学学报,2006,34(4)：470-472.
[15]潘丽静,郭鹏江,白美利.ρ合误差下回归权函数估计的强相合性[J].西北大学学报,2010,40(5)：756-759.
[16) Kolmogorov A. N., Rozanov U.A.. On the strong mixing conditions for stationary gaussion process[J]. Theor. Probab. Appl.1960,5:204-208.
[17]Rosenblatt, M.. Markov processes, structure and asymptotic behavior[M]. New York: Springer-Verlag,1971,216-217.
[18]Bradley, R. C. Equivalent mixing conditions for markov chains[J]. Statist. Probab. Letters, 1999,41(1999):97-99.
[19]Joag-Dev K., Proschan F.. Negative association of random variables with application[J]. Ann Statist,1983,11:286-295.
[20]邵启满.关于ρ混合序列的完全收敛性[J].数学学报,1989,32(3)：372-393.
[21 ]杨文志.NOD序列的指数不等式及其应用[D].安徽：安徽大学,2010.
[22]罗中德.ρ混合序列下估计的渐近性质[D].广西：广西师范大学,2010.
[23]Yang S. C. Uniformly asymptotic normality of regression weighted estimator for negatively associated sample[J]. Statist. Probab. Lett,2003,62,101-110.
[24]Kolmogorov A. N., Rozanov U. A.. On the strong mixing conditions for stationary gaussion process[J]. Theor. Probab. Appl,1960,5,204-208.
[25]Peter J., Brockwell, Richard A. Davis.. Time series:therory and meth-ods[M]. Heidelberg: Springer Verlag,1998.
[26]王学军.弱鞅和三类相依序列的概率不等式及极限定理[D].安徽：安徽大学,2010.
[27]杨善朝,李永明.强混合样本下回归加权估计的一致渐进正态性[J].数学学报,2006(5)：1164-1170
[28]李军,杨善朝.NA样本下固定设计回归估计的渐进正态性[J].工程数学学报,2004,21(1)：97-102.
[29]Yang S.C.. Moment bounds for strong mixing sequences and their application [J]. Mathemat-ical Rasearch and Exposition,2000,20(3):349-359.
[30]George G. Roussas.. Fixed design regression for time series:asymptotic normality[J]. Jour-nal of Multivariate Analysis,1992,4:262-291.
[31]胡舒合.固定设计下的非参数多元回归[J].科学通报,1989,16：1275-1276.
[32]苏淳,赵林城,王岳宝.NA序列的矩不等式与弱收敛[J].中国科学(A),1996,26(12)：1091-1099.
[33]林正炎,陆传荣,苏中根.概率极限理论基础[M].北京：高等教育出版社,1999.
[34]王成名,余鑫晖.应用概率统计[M].广西师范大学出版社,2003.