基于NQD样本密度函数估计的渐近性质研究
|
摘要
本文主要研究了基于两两NQD样本的密度函数估计问题,得到了两两NQD样本下最近邻估计和核估计的大样本性质,如相合性、渐近正态性及收敛速度等。从而推广了独立同分布和其它相依情形下的相关结果。
全文共分四部分。
第一部分介绍了密度函数估计的背景、意义、方法及本文的主要成果。
第二部分介绍了基于两两NQD样本最近邻密度估计的相合性并讨论了它们的收敛速度。
第三部分重点讨论了基于两两NQD样本密度函数核估计的强相合、一致强相合及r-阶矩相合。
第四部分讨论了基于两两NQD样本密度函数核估计的渐近正态性。
This paper is mainly concerned with the density function estimation problems under pairwises NQD samples, based on which we got the consistency, asymptotic property as well as the consistency rates and other large sample properties of the nearest neighbor estimation and kernel estimation. All the studies together have promoted the concerned results in i.i.d case and other associations
This paper consists of four parts.
In the first section, we introduced some background information, significance, the way of density function estimation and the chief results of this paper.
In the second section, we introduced the consistency of nearest neighbor density estimation under pairwises NQD samples. In addition, we also discussed its consistency rates.
In the third section, we focused our attention on the density kernel estimation under pairwises NQD samples and mainly talked about its strong consistency, uniformly strong consistency and the consistency in r-th order mean.
In the last section, we investigated the asymptotic normality of density function kernel estimation under pairwises NQD samples.
全文共分四部分。
第一部分介绍了密度函数估计的背景、意义、方法及本文的主要成果。
第二部分介绍了基于两两NQD样本最近邻密度估计的相合性并讨论了它们的收敛速度。
第三部分重点讨论了基于两两NQD样本密度函数核估计的强相合、一致强相合及r-阶矩相合。
第四部分讨论了基于两两NQD样本密度函数核估计的渐近正态性。
This paper is mainly concerned with the density function estimation problems under pairwises NQD samples, based on which we got the consistency, asymptotic property as well as the consistency rates and other large sample properties of the nearest neighbor estimation and kernel estimation. All the studies together have promoted the concerned results in i.i.d case and other associations
This paper consists of four parts.
In the first section, we introduced some background information, significance, the way of density function estimation and the chief results of this paper.
In the second section, we introduced the consistency of nearest neighbor density estimation under pairwises NQD samples. In addition, we also discussed its consistency rates.
In the third section, we focused our attention on the density kernel estimation under pairwises NQD samples and mainly talked about its strong consistency, uniformly strong consistency and the consistency in r-th order mean.
In the last section, we investigated the asymptotic normality of density function kernel estimation under pairwises NQD samples.
引文
[1]Rosenblatt,M.Remarks on some nonparametric estimates of a density function,Ann.Math Statist,1956,27:832-837.
[2]Parzen,E.On estimation of a probability density function and mode Ann.Math.Statist,1962,33:1065-76.
[3]Loftsgarden DO,Quesenberry CD.A Nonparametric estimator of a multivatiate density function.Ann.Statist,1965,36:1049-1051.
[4]Wagner T J.Strong consistency of anonparametic estimate of a density function.IEE E Trans Systems Man Cyberne 1973,3:289-290.
[5]Cheng X,R.Convergence rates for nearest neighbor density estimator,Science of China,1980,12:1419-1428.
[6]Yang Shanchao.The rate strong consistency of nearest neighbor density for independent samples[J],Chinese Journal of Applied Probability and Statistics,2002,18:405-408
[7]Boente G,Fraiman R.Consistency of a nonparametric estimate of a density function for dependent variables[J].Multivariate Analtsis,1988,25:90-99
[8]Chai G X.Consistency of nearest neighbor density estimator of stationary processes[J].Acta Mathemati Sinica,1989,2(3):423-432(in Chinese).
[9]柴根象.α-混合序列的最近邻密度估计[J].同济大学学报:2003.5:605-609
[10]Yang 5.C.Consistency of nearest neighbor density function for pairwises NA sequence.Act a.Math Application[J],2003,26(3):385-395.
[11]Liu Yanyan,Zhang Yanli.Consistency of nearest neighbor estimator of density function for pairwises negative quadrant dependent sequences[J].Wuhan Univ.2006,52(1):013-016
[12]熊丹.独立样本核密度估计的r阶均方相合性及收敛速度[J].数学杂志:2004.23:303-306
[13]林正炎.相依样本情形时密度的核估计[J],科学通报,1983,28(12):709-713.
[14]李军.φ-混合序列密度估计的强相合性[J].广西师大学报:1999.17:51-55
[15]Wei Laisheng,The consistencies for the Kernel-Type Density Estimation In the Case of Na Samples[J].J.Sys.Sci.&Math.Scis.2001,21(1):79-87.
[16]Wen Zhichen,Yang Shanchao.Consistency of the Density Kernel Estimator for Negatively and Positively Associated Samples.Journal of Mathematical Study[J],2002,35(3):309-319.
[17]杨善朝.NA样本最近邻密度估计的相合性[J].应用数学学报,2003(7):385-395.
[18]凌能祥.线性过程误差下概率密度函数核估计的均方相合性[J].纯粹数学与应用数学,2004,20(3):99-102.
[19]杨善朝.φ-混合样本密度估计的渐近正态性[J].广西师范大学学报,1996,14(1):18-21
[20]Pan,J.M,On the convergence rates in the central limit theorem for negative associated sequences,Chinese Journal of Appl.Prob.and.Stat,1997(13),183-192.
[21]Campos VSM.Kernel density estimation:the general case[J].Statistics and Prob Letters,2001,55(2):173-180
[22]George G.Roussas,asymptotic normality of the kernel estimate of aprobability density function under association.Statistics and Prob Letters,2000,50:1-12
[23]李永明,杨善朝.NA随机变量的递归密度核估计的渐近正态性.应用概率统计2003(19):383-393
[24]李永明,杨善朝.NA样本密度函数估计一致渐近正态性的收敛速度[J].数学物理学报,2005,25A(5).643-651.
[25]孙志宾.相依样本下概率密度函数估计的渐近正态性[J].数学的实践与认识,2001(31):727-731
[26]Lehamann PL.Some concepts of dependence[J].Ann.Math.Statistics,1966.43:1137-1153.
[27]吴群英.混合序列的概率极限理论[M]北京:科学出版社,2006.170-178.
[28]Matula P.A note on the almost sure convergence of sums of negatively dependent random variables[J].Statist.Probab.Lett.1992,15(3):209-213.
[29]王岳宝,苏淳.关于两两NQD列的若干极限性质[J].应用数学学报,1998,21(3):404-414.
[30]吴群英.两两NQD列的广义Jamison型加权的强收敛性[J].数学研究,2001,34(4):386-393.
[31]吴群英.两两NQD列的收敛性质[J].数学学报,2002,45(3):617-624.
[32]JamisonB,Orey S and Pruitt W.Convergence of weighted of independent [J].Random variables.Z.Wahrsch Verb Gebiete,1965(4):40-44
[33]万成高.两两NQD列的大数定律和完全收敛性[J].应用数学学报,2005,28(2):253-261.
[34]王志刚.两两NQD列弱大数定律及L~p收敛性[J],海南大学学报,2006.24(3):230-233
[35]刘莉.NQD样本下部分线性模型中估计的强相合性[J].湖北大学学报(自然科学版),2004,26(4):290-302.
[36]陈希孺,柴根象.非参数统计教程[M]上海:华东师范大学出版社,1993.261-270.
[37]安军.两两NOD序列部分和的不等式及弱大数律[J].重庆工商大学学报(自然科学版)2004,21(3):209-212.
[38]Bulinski,A.V.1996.On the convergence rates in the CLT for positive and Negatively dependent random fields.In:ibraginmov,I.A
[39]Jeffrey P.Kharoufeh & Konstadinos G.Goulias.Nonparametric Identification of Daily Activity Durations Using Kernel Density Estimators.Transportation Research Part B,2002,Vo 1.3 6,59-82.
[40]M.J.Ba xter&C.C.Beardah.Some Archaeological Application of Kernel Density Estimate.Journal of ArchaeologicalScience,1997,No.24,347-354.
[41]Kyle Crammer.Kernel Estimate in High-energy Physics.Computer Physics Communications,2001,Vol.1 36,198-207.
[42]Joachim Inkmann Misspecified heteroskedasticity in the panel probit model:A small sample comparison of GML and SML estimators.Journal of Econometrics,2000,97:227-259
[43]Charles Taylor.Classification and Kernel Density Estimate.Vistas in Astronomy,1997,Vol.41:411-417.
[44]Breunig,RobertV.Density Estimation for Clustered Data.Econometric Rev.2001,Vol.20,No.3:353-367
[45]Cai Z,RoussasG G.Weak convergence for a smooth estimator of a distribution function under association[J].Stochastic Anal.Appl.1999,17:145-168.
[46]张杰,杨振海.基于梯型密度函数的连续分布随机数近似生成方法[J].高校应用数学学报A辑,2004,19(1):118-124
[47]Yuan Ming,Su Chun,Weak convergence for empirical processes of negatively associated sequences,Chinese Journal of Appl.Prob.and.Stat,2000(16):45-56
[48]胡舒合,陈明华,方红.一般形式的密度估计.系统科学与数学,2005,25(2):228-236.
[49]任哲,陈华明,胡舒合.NA样本下一般形式的密度估计[J].大学数学,2005,21(4):41-44.
[50]林正炎,白志东.概率不等式[M]北京:科学出版社,2006.46-56.
[2]Parzen,E.On estimation of a probability density function and mode Ann.Math.Statist,1962,33:1065-76.
[3]Loftsgarden DO,Quesenberry CD.A Nonparametric estimator of a multivatiate density function.Ann.Statist,1965,36:1049-1051.
[4]Wagner T J.Strong consistency of anonparametic estimate of a density function.IEE E Trans Systems Man Cyberne 1973,3:289-290.
[5]Cheng X,R.Convergence rates for nearest neighbor density estimator,Science of China,1980,12:1419-1428.
[6]Yang Shanchao.The rate strong consistency of nearest neighbor density for independent samples[J],Chinese Journal of Applied Probability and Statistics,2002,18:405-408
[7]Boente G,Fraiman R.Consistency of a nonparametric estimate of a density function for dependent variables[J].Multivariate Analtsis,1988,25:90-99
[8]Chai G X.Consistency of nearest neighbor density estimator of stationary processes[J].Acta Mathemati Sinica,1989,2(3):423-432(in Chinese).
[9]柴根象.α-混合序列的最近邻密度估计[J].同济大学学报:2003.5:605-609
[10]Yang 5.C.Consistency of nearest neighbor density function for pairwises NA sequence.Act a.Math Application[J],2003,26(3):385-395.
[11]Liu Yanyan,Zhang Yanli.Consistency of nearest neighbor estimator of density function for pairwises negative quadrant dependent sequences[J].Wuhan Univ.2006,52(1):013-016
[12]熊丹.独立样本核密度估计的r阶均方相合性及收敛速度[J].数学杂志:2004.23:303-306
[13]林正炎.相依样本情形时密度的核估计[J],科学通报,1983,28(12):709-713.
[14]李军.φ-混合序列密度估计的强相合性[J].广西师大学报:1999.17:51-55
[15]Wei Laisheng,The consistencies for the Kernel-Type Density Estimation In the Case of Na Samples[J].J.Sys.Sci.&Math.Scis.2001,21(1):79-87.
[16]Wen Zhichen,Yang Shanchao.Consistency of the Density Kernel Estimator for Negatively and Positively Associated Samples.Journal of Mathematical Study[J],2002,35(3):309-319.
[17]杨善朝.NA样本最近邻密度估计的相合性[J].应用数学学报,2003(7):385-395.
[18]凌能祥.线性过程误差下概率密度函数核估计的均方相合性[J].纯粹数学与应用数学,2004,20(3):99-102.
[19]杨善朝.φ-混合样本密度估计的渐近正态性[J].广西师范大学学报,1996,14(1):18-21
[20]Pan,J.M,On the convergence rates in the central limit theorem for negative associated sequences,Chinese Journal of Appl.Prob.and.Stat,1997(13),183-192.
[21]Campos VSM.Kernel density estimation:the general case[J].Statistics and Prob Letters,2001,55(2):173-180
[22]George G.Roussas,asymptotic normality of the kernel estimate of aprobability density function under association.Statistics and Prob Letters,2000,50:1-12
[23]李永明,杨善朝.NA随机变量的递归密度核估计的渐近正态性.应用概率统计2003(19):383-393
[24]李永明,杨善朝.NA样本密度函数估计一致渐近正态性的收敛速度[J].数学物理学报,2005,25A(5).643-651.
[25]孙志宾.相依样本下概率密度函数估计的渐近正态性[J].数学的实践与认识,2001(31):727-731
[26]Lehamann PL.Some concepts of dependence[J].Ann.Math.Statistics,1966.43:1137-1153.
[27]吴群英.混合序列的概率极限理论[M]北京:科学出版社,2006.170-178.
[28]Matula P.A note on the almost sure convergence of sums of negatively dependent random variables[J].Statist.Probab.Lett.1992,15(3):209-213.
[29]王岳宝,苏淳.关于两两NQD列的若干极限性质[J].应用数学学报,1998,21(3):404-414.
[30]吴群英.两两NQD列的广义Jamison型加权的强收敛性[J].数学研究,2001,34(4):386-393.
[31]吴群英.两两NQD列的收敛性质[J].数学学报,2002,45(3):617-624.
[32]JamisonB,Orey S and Pruitt W.Convergence of weighted of independent [J].Random variables.Z.Wahrsch Verb Gebiete,1965(4):40-44
[33]万成高.两两NQD列的大数定律和完全收敛性[J].应用数学学报,2005,28(2):253-261.
[34]王志刚.两两NQD列弱大数定律及L~p收敛性[J],海南大学学报,2006.24(3):230-233
[35]刘莉.NQD样本下部分线性模型中估计的强相合性[J].湖北大学学报(自然科学版),2004,26(4):290-302.
[36]陈希孺,柴根象.非参数统计教程[M]上海:华东师范大学出版社,1993.261-270.
[37]安军.两两NOD序列部分和的不等式及弱大数律[J].重庆工商大学学报(自然科学版)2004,21(3):209-212.
[38]Bulinski,A.V.1996.On the convergence rates in the CLT for positive and Negatively dependent random fields.In:ibraginmov,I.A
[39]Jeffrey P.Kharoufeh & Konstadinos G.Goulias.Nonparametric Identification of Daily Activity Durations Using Kernel Density Estimators.Transportation Research Part B,2002,Vo 1.3 6,59-82.
[40]M.J.Ba xter&C.C.Beardah.Some Archaeological Application of Kernel Density Estimate.Journal of ArchaeologicalScience,1997,No.24,347-354.
[41]Kyle Crammer.Kernel Estimate in High-energy Physics.Computer Physics Communications,2001,Vol.1 36,198-207.
[42]Joachim Inkmann Misspecified heteroskedasticity in the panel probit model:A small sample comparison of GML and SML estimators.Journal of Econometrics,2000,97:227-259
[43]Charles Taylor.Classification and Kernel Density Estimate.Vistas in Astronomy,1997,Vol.41:411-417.
[44]Breunig,RobertV.Density Estimation for Clustered Data.Econometric Rev.2001,Vol.20,No.3:353-367
[45]Cai Z,RoussasG G.Weak convergence for a smooth estimator of a distribution function under association[J].Stochastic Anal.Appl.1999,17:145-168.
[46]张杰,杨振海.基于梯型密度函数的连续分布随机数近似生成方法[J].高校应用数学学报A辑,2004,19(1):118-124
[47]Yuan Ming,Su Chun,Weak convergence for empirical processes of negatively associated sequences,Chinese Journal of Appl.Prob.and.Stat,2000(16):45-56
[48]胡舒合,陈明华,方红.一般形式的密度估计.系统科学与数学,2005,25(2):228-236.
[49]任哲,陈华明,胡舒合.NA样本下一般形式的密度估计[J].大学数学,2005,21(4):41-44.
[50]林正炎,白志东.概率不等式[M]北京:科学出版社,2006.46-56.