弱误差半参数和非参数回归模型估计的相合性
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摘要
近些年来,φ混合序列和ψ混合序列及NOD序列等相依序列的理论研究得到了充分的发展,特别是一些重要的不等式,如Bernstein不等式,Rosenthal型不等式等,这促使了这些序列在统计领域得到了很好的发展.在误差序列为上述相依序列下,本文主要致力于研究半参数和非参数回归模型估计的相合性问题.
本文的第二章研究了半参数回归模型Y(j)(xin,tin)=tinβ+g(xin)+e(j)(xin),1≤j≤m,1≤i≤n,综合了最小二乘法和权函数的估计方法,定义了β和g的估计量βm,n和gm,n(x)通过截尾的方法,利用φ混合序列和ψ混合序列的矩不等式以及凸函数的性质,在φ混合和ψ混合误差及其他条件下证明了它们的r(r>2)阶矩相合性和强相合性,推广了胡舒合(1997)的相应结果.
本文的第三章考虑了非参数回归模型Yi=g(xi)+εi,1≤i≤n,定义了未知函数g(x)的估计量gn(x),在通常假设的条件下,证明了在误差为NOD序列下g(x)估计量的r(r>1)阶矩相合性,强相合性及完全收敛性.同时,在一致情形的假设下得到了g(x)估计量的一致的矩相合性和强相合性.由于独立序列和NA序列是特殊的NOD序列,我们所得结果推广了非参数回归模型在误差为独立和NA情形下的相应的结果.
In recent years.the theory of dependent sequences such asφmixing sequence.Ψmix-ing sequence and NOD sequence and so on, has been sufficiently developed. Especially sev-eral important inequalities are obtained, for instance, Bernstein inequality, Rosenthal'type inequality, and so forth, which greatly improves the development of the applications in statistical fields. In this paper, the consistency of the estimators for semiparametric and nonparametric regressional models are estabilished under the dependent errors above.
In Chapter 2, the semiparametric regression model Y(j)(xin,tin)=tinβ+g(xin)+ e(j)(xin),1≤j≤m,1≤i≤n is considered. Based on the methods of least squares and weight function, the estimatorsβm,n and gm,n(x) forβand g are defined, respectively. By using the truncating method, the moment inequalities forφmixing sequence andΨmixing sequence and the properties of the convex functions, we obtain the r-th moment consistency and the strong consistency for these estimators under some mild conditions, which generalize the correponding results of Hu(1997).
In Chapter 3, the nonparametric regression model Yi=g(xi)+εi for i=1,2,…,n is discussed.The estimator gn(x) of the unknown function g(x) is defined. For NOD error sequence, under some mild conditions, we investigate the r-th moment consistency, the strong consistency and almostly complete convergence for the estimator. Moreover, under some uniform assumptions,the uniform moment consistency and the uniform strong con-sistency are also obtained. Since independent random variables and negatively associated random variables are the cases of NOD random variables, the results of independent and negatively associated sequences are generalized by the ones that we obtain.
本文的第二章研究了半参数回归模型Y(j)(xin,tin)=tinβ+g(xin)+e(j)(xin),1≤j≤m,1≤i≤n,综合了最小二乘法和权函数的估计方法,定义了β和g的估计量βm,n和gm,n(x)通过截尾的方法,利用φ混合序列和ψ混合序列的矩不等式以及凸函数的性质,在φ混合和ψ混合误差及其他条件下证明了它们的r(r>2)阶矩相合性和强相合性,推广了胡舒合(1997)的相应结果.
本文的第三章考虑了非参数回归模型Yi=g(xi)+εi,1≤i≤n,定义了未知函数g(x)的估计量gn(x),在通常假设的条件下,证明了在误差为NOD序列下g(x)估计量的r(r>1)阶矩相合性,强相合性及完全收敛性.同时,在一致情形的假设下得到了g(x)估计量的一致的矩相合性和强相合性.由于独立序列和NA序列是特殊的NOD序列,我们所得结果推广了非参数回归模型在误差为独立和NA情形下的相应的结果.
In recent years.the theory of dependent sequences such asφmixing sequence.Ψmix-ing sequence and NOD sequence and so on, has been sufficiently developed. Especially sev-eral important inequalities are obtained, for instance, Bernstein inequality, Rosenthal'type inequality, and so forth, which greatly improves the development of the applications in statistical fields. In this paper, the consistency of the estimators for semiparametric and nonparametric regressional models are estabilished under the dependent errors above.
In Chapter 2, the semiparametric regression model Y(j)(xin,tin)=tinβ+g(xin)+ e(j)(xin),1≤j≤m,1≤i≤n is considered. Based on the methods of least squares and weight function, the estimatorsβm,n and gm,n(x) forβand g are defined, respectively. By using the truncating method, the moment inequalities forφmixing sequence andΨmixing sequence and the properties of the convex functions, we obtain the r-th moment consistency and the strong consistency for these estimators under some mild conditions, which generalize the correponding results of Hu(1997).
In Chapter 3, the nonparametric regression model Yi=g(xi)+εi for i=1,2,…,n is discussed.The estimator gn(x) of the unknown function g(x) is defined. For NOD error sequence, under some mild conditions, we investigate the r-th moment consistency, the strong consistency and almostly complete convergence for the estimator. Moreover, under some uniform assumptions,the uniform moment consistency and the uniform strong con-sistency are also obtained. Since independent random variables and negatively associated random variables are the cases of NOD random variables, the results of independent and negatively associated sequences are generalized by the ones that we obtain.
引文
[1]胡舒合,潘光明,高启兵.误差为线性过程时回归模型的估计问题[J].高校应用数学学报A辑,2003.18(01)：81-90.
[2]白美利.误差为线性过程时半参数模型的估计[J].纯粹数学与应用数学,2010,26(1)：171-176.
[3]凌能祥.线性过程误差下的半参数回归模型[J].安徽大学学报(自然科学版),2004,28(04)：15-18.
[4]胡舒合,张林松,王吟,沈燕,方红.Lp-混合误差下回归模型估计量的平均相合性[J].安徽大学学报(自然科学版),2004,28(01)：1-9.
[5]周建新,胡宏昌.半参数回归模型小波估计的矩相合性[J].数学的实践与认识,2010,40(23)：170-176.
[6]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(自然科学版),2009,32(01)：136-138.
[7]胡宏昌.长相依半参数回归模型的小波估计[J].数学学报,2009,52(04)：641-650.
[8]李永明,韩龙生.NA序列半参数回归模型小波估计的强相合性[J].数学的实践与认识,2007,37(20)：47-52.
[9]刘强,薛留根.混合误差下半参数回归模型小波估计的强相合性[J].数学的实践与认识,2008,38(10)：97-101.
[10]胡宏昌,胡迪鹤.半参数回归模型小波估计的强相合性[J].数学学报,2006,49(06)：1417-1424.
[11]李琪琪,韦来生.半参数回归模型中参数的Bayes估计[J].中国科学技术大学学报,2010,40(09)：881-886.
[12]周兴才,胡舒合.NA样本半参数回归模型估计的矩相合性[J].纯粹数学与应用数学,2010,26(02)：262-269.
[13]胡舒合.相依误差下线性模型参数估计的渐近正态性[J].科学通报,1998,43(23)：2489-2492.
[14]胡舒合.一类半参数回归模型的估计问题[J].数学物理学报,1999,19(5)：541-549.
[15]潘光明,胡舒合,方利宝,程正东.半参数回归模型估计的平均相合性[J].数学物理学报,2003,23A(05)：598-606.
[16]陈明华,任哲,胡舒合.部分线性模型中估计的强相合性[J].数学学报,1998,41(02)：429-438.
[17]任哲,陈明华.NA样本下部分线性模型中估计的强相合性[J].应用概率统计,2002,18(01)：60-66.
[18]朱春浩,孙光辉.随机截断下NA样本半参数回归模型中的相合估计[J].数学杂志,2007,27(03)：327-332.
[19]田萍,马国锋.一类纵向数据半参数模型中的强相合估计[J].数理统计与管理,2008.27(05)：864-868.
[20]胡宏昌,徐侃,陈琴.半参数回归模型拟极大似然估计的弱相合性[J].工程数学学报,2008,25(06)：1081-1086.
[21]田萍,薛留根.纵向数据半参数回归模型估计的强相合性[J].工程数学学报,2006,23(2)：369-372.
[22]李国亮,刘禄勤.误差为鞅差序列的部分线性模型中估计的强相合性[J].2007,27A(05)：788-801.
[23]Dobrushin, R. L.The central limit theorem for non-stationary Markov china[J]. Theory of Probability and its Applications,1956,1:72-88.
[24]Kolmogorov, A. N., Rozanov, U.A.On the strong mixing conditions of a stationary Gaussian process[J]. Theory of Probability and its Applications,1960,2:222-227.
[25]Iosifescu, J. Limit theorem for φ-mixing sequences[J]. A survey. Proc.5th Conf. on Probab. Theory, Sept.1-6,1974, Brasov, Romania(Editura Acad. R. S. R., Bucure-sti),1977:51-57.
[26]Herrndorf, N. The invariance principle for φ-mixing sequences[J]. Z.Wahrsch. verw Gebiete, 1983,63:97-108.
[27]Peligrad, M. An invariance principle for φ-mixing sequences [J]. Annal of Probability,1985, 13:1304-1313.
[28]Utev, S.S. On the CLT for the series scheme of random variables with the φ-mixing[J]. Theory of Probability and its Applications,1990,35:110-117.
[29]Chen, D. Q. A uniform central limit theorem for non-uniform φ-mixing random fields[J]. Annal of Probability,1991,19:636-649.
[30]Shen Aiting. Some New Convergence Properties for φmixing Random Variable Sequences[J]. Matheatica Applicata,2011,24(1),1-9.
[31]杨善朝.混合序列加权和的强收敛性[J].系统科学与数学,1995,15(3)：254-265.
[32]Stout W. F. Almost Sure Convergence[M]. New York:Academic Press,1974,13.
[33]Bozorgnia A., Patterson R.F., Taylor R.L. Limit theorems for dependent random vari-ables[C]. Proceedings of the first world congress on World congress of nonlinear ana- lysts'92(II). Berlin:Walter de Grutyer,1996:1639-1650.
[34]Asadian N., Fakoor V., Bozorgnia A. Rosenthal's Type Inequalities for Negatively Orthant Dependent Random Variables[J]. Journal of the Iranian Statistical Society,2006,5:69-75.
[35]Blum J.R., Hanson D.L., Koopmans L. On the strong law of large numbers for a class of stochastic processes[J]. Z.Wahrsch.verw.Gebiete,1963,2:1-11.
[36]Joag-Dev K., Proschan F.Negative association of random variables with applications[J]. An-nal of Statistics,1983,11(1):286-295.
[37]Georgiev A. A. Local properties of function fitting estimates with applications to system identification, in:W.Grossmann et al.(Eds.), Mathematical Statistics and Applications, Pro-ceeding 4th Pannonian Sump. Math.Statist.4-10, September,1983, Bad Tatzmannsdorf, Austria, Reidel, Dordrecht,1985, pp.141-151.
[38]Georgiev A. A. Greblicki W. Nonparametric function recovering from noisy observation[J]. Journal of Statistical Planning and Inference,1986,13:1-14.
[39]Engle R. F., Granger C. W. J, Rice J., Weiss. Nonparametric estimates of the relation between weather and electricity sales[J]. Journal of the American Statistical Association, 1986,81:310-320.
[40]Muller H. G. Weak and universal consistency of moving weighted averages[J]. Periodica Mathematica Hungarica,1987,18(3):241-250.
[41]Georgiev A. A. Consistent nonparametric multiple regression:the fixed design case[J].Jour-nal of Multivariate Analysis,1988,25(1):100-110.
[42]Roussas G. G. Consistent regression estimation with fixed design points under dependence conditions[J]. Statistics and Probability Letters.1989,8:41-50.
[43]Fan Y. Consistent nonparametric multiple regression for dependent heterogeneous pro-cesses[J].Journal of Multivariate Analysis.1990,33(1):72-88.
[44]Fraiman R., Iribarren P. Nonparametric regression estimation in models with weak error's structure[J].Journal of Multivariate Analysis,1991,37(2):180-196.
[45]洪圣岩.一类半参数回归模型的估计理论[J].中国科学,1991,12(A)：1258-1272.
[46]Bryc W., Smolenski W. Moment condition for almost sure convergence of weakly correlated random variables[J]. Proceedings of American Mathematical Society.1993,119(2):629-635.
[47]王启华.随机截断下半参数回归模型中的相合估计[J].中国科学,1995,8(A)：818-832.
[48]胡舒合.一类新的半参数回归模型中的相合估计[J].数学学报,1997,40(4)：527-536.
[49]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.
[50]陈明华.固定设计下半参数回归模型估计的相合性[J].高校应用数学学报,1998,13A(3)：301-310.
[51]吴本忠.混合误差半参数回归模型估计的强相合性[J].应用数学,1998,11(3)：27-31.
[52]杨善朝.一类随机变量部分和的矩不等式及其应用[J].科学通报,1998,43(17)：1823-1827.
[53]杨善朝.基于鞅序列非参数回归权函数的估计[J].系统科学与数学,1999,19(01)：79-85.
[54]胡舒合,胡晓鹏,潘光明.Lp混合误差下线性与非参数回归模型估计量的平均相合性[J].应用数学学报,2003,26(04)：756-759.
[55]李军,杨善朝.相协样本半参数回归模型估计的矩相合性[J].应用数学,2004,17(2)：257-262.
[56]Liang H.Y., Jing B.Y. Asymptotic properties for estimates of nonparametric regression mod-els based on negatively associated sequences[J].Journal of Multivariate Analysis,2005,95(2): 227-245.
[2]白美利.误差为线性过程时半参数模型的估计[J].纯粹数学与应用数学,2010,26(1)：171-176.
[3]凌能祥.线性过程误差下的半参数回归模型[J].安徽大学学报(自然科学版),2004,28(04)：15-18.
[4]胡舒合,张林松,王吟,沈燕,方红.Lp-混合误差下回归模型估计量的平均相合性[J].安徽大学学报(自然科学版),2004,28(01)：1-9.
[5]周建新,胡宏昌.半参数回归模型小波估计的矩相合性[J].数学的实践与认识,2010,40(23)：170-176.
[6]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(自然科学版),2009,32(01)：136-138.
[7]胡宏昌.长相依半参数回归模型的小波估计[J].数学学报,2009,52(04)：641-650.
[8]李永明,韩龙生.NA序列半参数回归模型小波估计的强相合性[J].数学的实践与认识,2007,37(20)：47-52.
[9]刘强,薛留根.混合误差下半参数回归模型小波估计的强相合性[J].数学的实践与认识,2008,38(10)：97-101.
[10]胡宏昌,胡迪鹤.半参数回归模型小波估计的强相合性[J].数学学报,2006,49(06)：1417-1424.
[11]李琪琪,韦来生.半参数回归模型中参数的Bayes估计[J].中国科学技术大学学报,2010,40(09)：881-886.
[12]周兴才,胡舒合.NA样本半参数回归模型估计的矩相合性[J].纯粹数学与应用数学,2010,26(02)：262-269.
[13]胡舒合.相依误差下线性模型参数估计的渐近正态性[J].科学通报,1998,43(23)：2489-2492.
[14]胡舒合.一类半参数回归模型的估计问题[J].数学物理学报,1999,19(5)：541-549.
[15]潘光明,胡舒合,方利宝,程正东.半参数回归模型估计的平均相合性[J].数学物理学报,2003,23A(05)：598-606.
[16]陈明华,任哲,胡舒合.部分线性模型中估计的强相合性[J].数学学报,1998,41(02)：429-438.
[17]任哲,陈明华.NA样本下部分线性模型中估计的强相合性[J].应用概率统计,2002,18(01)：60-66.
[18]朱春浩,孙光辉.随机截断下NA样本半参数回归模型中的相合估计[J].数学杂志,2007,27(03)：327-332.
[19]田萍,马国锋.一类纵向数据半参数模型中的强相合估计[J].数理统计与管理,2008.27(05)：864-868.
[20]胡宏昌,徐侃,陈琴.半参数回归模型拟极大似然估计的弱相合性[J].工程数学学报,2008,25(06)：1081-1086.
[21]田萍,薛留根.纵向数据半参数回归模型估计的强相合性[J].工程数学学报,2006,23(2)：369-372.
[22]李国亮,刘禄勤.误差为鞅差序列的部分线性模型中估计的强相合性[J].2007,27A(05)：788-801.
[23]Dobrushin, R. L.The central limit theorem for non-stationary Markov china[J]. Theory of Probability and its Applications,1956,1:72-88.
[24]Kolmogorov, A. N., Rozanov, U.A.On the strong mixing conditions of a stationary Gaussian process[J]. Theory of Probability and its Applications,1960,2:222-227.
[25]Iosifescu, J. Limit theorem for φ-mixing sequences[J]. A survey. Proc.5th Conf. on Probab. Theory, Sept.1-6,1974, Brasov, Romania(Editura Acad. R. S. R., Bucure-sti),1977:51-57.
[26]Herrndorf, N. The invariance principle for φ-mixing sequences[J]. Z.Wahrsch. verw Gebiete, 1983,63:97-108.
[27]Peligrad, M. An invariance principle for φ-mixing sequences [J]. Annal of Probability,1985, 13:1304-1313.
[28]Utev, S.S. On the CLT for the series scheme of random variables with the φ-mixing[J]. Theory of Probability and its Applications,1990,35:110-117.
[29]Chen, D. Q. A uniform central limit theorem for non-uniform φ-mixing random fields[J]. Annal of Probability,1991,19:636-649.
[30]Shen Aiting. Some New Convergence Properties for φmixing Random Variable Sequences[J]. Matheatica Applicata,2011,24(1),1-9.
[31]杨善朝.混合序列加权和的强收敛性[J].系统科学与数学,1995,15(3)：254-265.
[32]Stout W. F. Almost Sure Convergence[M]. New York:Academic Press,1974,13.
[33]Bozorgnia A., Patterson R.F., Taylor R.L. Limit theorems for dependent random vari-ables[C]. Proceedings of the first world congress on World congress of nonlinear ana- lysts'92(II). Berlin:Walter de Grutyer,1996:1639-1650.
[34]Asadian N., Fakoor V., Bozorgnia A. Rosenthal's Type Inequalities for Negatively Orthant Dependent Random Variables[J]. Journal of the Iranian Statistical Society,2006,5:69-75.
[35]Blum J.R., Hanson D.L., Koopmans L. On the strong law of large numbers for a class of stochastic processes[J]. Z.Wahrsch.verw.Gebiete,1963,2:1-11.
[36]Joag-Dev K., Proschan F.Negative association of random variables with applications[J]. An-nal of Statistics,1983,11(1):286-295.
[37]Georgiev A. A. Local properties of function fitting estimates with applications to system identification, in:W.Grossmann et al.(Eds.), Mathematical Statistics and Applications, Pro-ceeding 4th Pannonian Sump. Math.Statist.4-10, September,1983, Bad Tatzmannsdorf, Austria, Reidel, Dordrecht,1985, pp.141-151.
[38]Georgiev A. A. Greblicki W. Nonparametric function recovering from noisy observation[J]. Journal of Statistical Planning and Inference,1986,13:1-14.
[39]Engle R. F., Granger C. W. J, Rice J., Weiss. Nonparametric estimates of the relation between weather and electricity sales[J]. Journal of the American Statistical Association, 1986,81:310-320.
[40]Muller H. G. Weak and universal consistency of moving weighted averages[J]. Periodica Mathematica Hungarica,1987,18(3):241-250.
[41]Georgiev A. A. Consistent nonparametric multiple regression:the fixed design case[J].Jour-nal of Multivariate Analysis,1988,25(1):100-110.
[42]Roussas G. G. Consistent regression estimation with fixed design points under dependence conditions[J]. Statistics and Probability Letters.1989,8:41-50.
[43]Fan Y. Consistent nonparametric multiple regression for dependent heterogeneous pro-cesses[J].Journal of Multivariate Analysis.1990,33(1):72-88.
[44]Fraiman R., Iribarren P. Nonparametric regression estimation in models with weak error's structure[J].Journal of Multivariate Analysis,1991,37(2):180-196.
[45]洪圣岩.一类半参数回归模型的估计理论[J].中国科学,1991,12(A)：1258-1272.
[46]Bryc W., Smolenski W. Moment condition for almost sure convergence of weakly correlated random variables[J]. Proceedings of American Mathematical Society.1993,119(2):629-635.
[47]王启华.随机截断下半参数回归模型中的相合估计[J].中国科学,1995,8(A)：818-832.
[48]胡舒合.一类新的半参数回归模型中的相合估计[J].数学学报,1997,40(4)：527-536.
[49]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.
[50]陈明华.固定设计下半参数回归模型估计的相合性[J].高校应用数学学报,1998,13A(3)：301-310.
[51]吴本忠.混合误差半参数回归模型估计的强相合性[J].应用数学,1998,11(3)：27-31.
[52]杨善朝.一类随机变量部分和的矩不等式及其应用[J].科学通报,1998,43(17)：1823-1827.
[53]杨善朝.基于鞅序列非参数回归权函数的估计[J].系统科学与数学,1999,19(01)：79-85.
[54]胡舒合,胡晓鹏,潘光明.Lp混合误差下线性与非参数回归模型估计量的平均相合性[J].应用数学学报,2003,26(04)：756-759.
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