NSD样本最近邻密度估计的相合性
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摘要
利用负超可加相依(NSD)序列的Bernstein不等式和Borel-Cantelli引理,给出NSD样本最近邻密度估计和失效率函数估计的(弱)强相合性、一致强相合性和(一致)强相合速度.
By using Bernstein inequality and Borel-Cantelli lemma of negatively superadditive dependent(NSD)sequence,we gave the(weak)strong consistency,uniform strong consistency and(uniform)strong consistency rate for the nearest neighbor estimator of density function and failure rate function of NSD samples.
By using Bernstein inequality and Borel-Cantelli lemma of negatively superadditive dependent(NSD)sequence,we gave the(weak)strong consistency,uniform strong consistency and(uniform)strong consistency rate for the nearest neighbor estimator of density function and failure rate function of NSD samples.
引文
[1]HU Taizhong.Negatively Superadditive Dependence of Random Variables with Applications[J].Chinese J Appl Probab Statist,2000,16(2):133-144.
[2]薛留根.现代非参数统计[M].北京:科学出版社,2015.(XUE Liugen.Modern Nonparametric Statistics[M].Beijing:Science Press,2015.)
[3]CHRISTOFIDES T C,VAGGELATOU E.A Connection between Supermodular Ordering and Positive/Negative Association[J].J Multivariate Anal,2004,88(1):138-151.
[4]SHEN Yan,WANG Xuejun,YANG Wenzhi,et al.Almost Sure Convergence Theorem and Strong Stability for Weighted Sums of NSD Random Variables[J].Acta Math Sin(Engl Ser),2013,29(4):743-756.
[5]WANG Xuejun,DENG Xin,ZHENG Lulu,et al.Complete Convergence for Arrays of Rowwise Negatively Superadditive-Dependent Random Variables and Its Applications[J].Statistics,2014,48(4):834-850.
[6]SHEN Aiting,WANG Xinghui.Kaplan-Meier Estimator and Hazard Estimator for Censored Negatively Superadditive Dependent Data[J].Statistics,2016,50(2):377-388.
[7]WU Yi,WANG Xuejun,HU Shuhe.Complete Convergence for Arrays of Rowwise Negatively SuperadditiveDependent Random Variables and Its Applications[J].Appl Math J Chinese Univ(Ser B),2016,31(4):439-457.
[8]余云彩,胡宏昌.NSD序列加权和的中心极限定理及其在EV回归模型中的应用[J].纯粹数学与应用数学,2016,32(5):525-535.(YU Yuncai,HU Hongchang.A CLT for Weighted Sums of NSD Random Sequences and Its Application in the EV Regression Model[J].Pure and Applied Mathematics,2016,32(5):525-535.)
[9]MENG Bing,WANG Dingcheng,WU Qunying.On the Strong Convergence for Weighted Sums of Negatively Superadditive Dependent Random Variables[J/OL].J Inequal Appl,2017-10-27.https://doi.org/10.1186/s13660-017-1530-9.
[10]ZHENG Lulu,WANG Xuejun,YANG Wenzhi.On the Strong Convergence for Weighted Sums of Negatively Superadditive Dependent Random Variables[J].Filomat,2017,31(2):295-308.
[11]NADERI H,AMINI M,BOZORGNIA A.On the Rate of Complete Convergence for Weighted Sums of NSDRandom Variables and an Application[J].Appl Math J Chinese Univ(Ser B),2017,32(3):270-280.
[12]YU Yuncai,HU Hongchang,LIU Ling,et al.M-Test in Linear Models with Negatively Superadditive Dependent Errors[J/OL].J Inequal Appl,2017-09-22.https://doi.org/10.1186/s13660-017-1509-6.
[13]WANG Xuejun,WU Yi,HU Shuhe.Strong and Weak Consistency of LS Estimators in the EV Regression Model with Negatively Superadditive-Dependent Errors[J].AStA Adv Stat Anal,2018,102(1):41-65.
[14]聂彩玲,李永明,应锐.负超可加相依样本密度函数核估计的相合性[J].应用数学,2018,31(2):457-462.(NIE Cailing,LI Yongming,YING Rui.Consistency of the Kernel Estimator of the Density Function for NSDSequence[J].Mathematica Applicata,2018,31(2):457-462.)
[15]陈希孺.最近邻密度估计的收敛速度[J].中国科学,1981,12:1419-1428.(CHEN Xiru.Rate of Convergence of Nearest Neighbor Estimator of Density Function[J].Scientia,1981,12:1419-1428.)
[16]柴根象.平稳序列最近邻密度估计的相合性[J].数学学报,1989,32(3):423-432.(CHAI Genxiang.The Consistency of Nearest Neighbor Estimator of Density Function for Stationary Sequences[J].Acta Mathematica Sinica,1989,32(3):423-432.)
[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,2003,26(3):385-395.(YANG Shanchao.Consistency of Nearest Neighbor Estimator of Density Function for Negatively Associated Samples[J].Acta Mathematicae Applicatae Sinica,2003,26(3):385-395.)
[18]刘永辉,吴群英.ND样本最近邻密度估计的相合性[J].吉林大学学报(理学版),2012,50(6):1141-1145.(LIU Yonghui,WU Qunying.Consistency of Nearest Neighbor Estimator of Density Function for Negatively Dependent Samples[J].Journal of Jilin University(Science Edition),2012,50(6):1141-1145.)
[19]兰冲锋,吴群英.END样本最近邻密度估计的一致强相合速度[J].吉林大学学报(理学版),2014,52(3):494-498.(LAN Chongfeng,WU Qunying.Uniformly Strong Consistency Rate of the Nearest Neighbor Estimator of Density Function for END Samples[J].Journal of Jilin University(Science Edition),2014,52(3):494-498.)
[20]李艺璇.END样本最近邻密度估计的相合性[J].湖北大学学报(自然科学版),2016,38(5):396-402.(LI Yixuan.Consistency of Nearest Neighbor Estimator of Density Function for Extended Negatively Dependent Samples[J].Journal of Hubei University(Natural Science),2016,38(5):396-402.)
[2]薛留根.现代非参数统计[M].北京:科学出版社,2015.(XUE Liugen.Modern Nonparametric Statistics[M].Beijing:Science Press,2015.)
[3]CHRISTOFIDES T C,VAGGELATOU E.A Connection between Supermodular Ordering and Positive/Negative Association[J].J Multivariate Anal,2004,88(1):138-151.
[4]SHEN Yan,WANG Xuejun,YANG Wenzhi,et al.Almost Sure Convergence Theorem and Strong Stability for Weighted Sums of NSD Random Variables[J].Acta Math Sin(Engl Ser),2013,29(4):743-756.
[5]WANG Xuejun,DENG Xin,ZHENG Lulu,et al.Complete Convergence for Arrays of Rowwise Negatively Superadditive-Dependent Random Variables and Its Applications[J].Statistics,2014,48(4):834-850.
[6]SHEN Aiting,WANG Xinghui.Kaplan-Meier Estimator and Hazard Estimator for Censored Negatively Superadditive Dependent Data[J].Statistics,2016,50(2):377-388.
[7]WU Yi,WANG Xuejun,HU Shuhe.Complete Convergence for Arrays of Rowwise Negatively SuperadditiveDependent Random Variables and Its Applications[J].Appl Math J Chinese Univ(Ser B),2016,31(4):439-457.
[8]余云彩,胡宏昌.NSD序列加权和的中心极限定理及其在EV回归模型中的应用[J].纯粹数学与应用数学,2016,32(5):525-535.(YU Yuncai,HU Hongchang.A CLT for Weighted Sums of NSD Random Sequences and Its Application in the EV Regression Model[J].Pure and Applied Mathematics,2016,32(5):525-535.)
[9]MENG Bing,WANG Dingcheng,WU Qunying.On the Strong Convergence for Weighted Sums of Negatively Superadditive Dependent Random Variables[J/OL].J Inequal Appl,2017-10-27.https://doi.org/10.1186/s13660-017-1530-9.
[10]ZHENG Lulu,WANG Xuejun,YANG Wenzhi.On the Strong Convergence for Weighted Sums of Negatively Superadditive Dependent Random Variables[J].Filomat,2017,31(2):295-308.
[11]NADERI H,AMINI M,BOZORGNIA A.On the Rate of Complete Convergence for Weighted Sums of NSDRandom Variables and an Application[J].Appl Math J Chinese Univ(Ser B),2017,32(3):270-280.
[12]YU Yuncai,HU Hongchang,LIU Ling,et al.M-Test in Linear Models with Negatively Superadditive Dependent Errors[J/OL].J Inequal Appl,2017-09-22.https://doi.org/10.1186/s13660-017-1509-6.
[13]WANG Xuejun,WU Yi,HU Shuhe.Strong and Weak Consistency of LS Estimators in the EV Regression Model with Negatively Superadditive-Dependent Errors[J].AStA Adv Stat Anal,2018,102(1):41-65.
[14]聂彩玲,李永明,应锐.负超可加相依样本密度函数核估计的相合性[J].应用数学,2018,31(2):457-462.(NIE Cailing,LI Yongming,YING Rui.Consistency of the Kernel Estimator of the Density Function for NSDSequence[J].Mathematica Applicata,2018,31(2):457-462.)
[15]陈希孺.最近邻密度估计的收敛速度[J].中国科学,1981,12:1419-1428.(CHEN Xiru.Rate of Convergence of Nearest Neighbor Estimator of Density Function[J].Scientia,1981,12:1419-1428.)
[16]柴根象.平稳序列最近邻密度估计的相合性[J].数学学报,1989,32(3):423-432.(CHAI Genxiang.The Consistency of Nearest Neighbor Estimator of Density Function for Stationary Sequences[J].Acta Mathematica Sinica,1989,32(3):423-432.)
[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,2003,26(3):385-395.(YANG Shanchao.Consistency of Nearest Neighbor Estimator of Density Function for Negatively Associated Samples[J].Acta Mathematicae Applicatae Sinica,2003,26(3):385-395.)
[18]刘永辉,吴群英.ND样本最近邻密度估计的相合性[J].吉林大学学报(理学版),2012,50(6):1141-1145.(LIU Yonghui,WU Qunying.Consistency of Nearest Neighbor Estimator of Density Function for Negatively Dependent Samples[J].Journal of Jilin University(Science Edition),2012,50(6):1141-1145.)
[19]兰冲锋,吴群英.END样本最近邻密度估计的一致强相合速度[J].吉林大学学报(理学版),2014,52(3):494-498.(LAN Chongfeng,WU Qunying.Uniformly Strong Consistency Rate of the Nearest Neighbor Estimator of Density Function for END Samples[J].Journal of Jilin University(Science Edition),2014,52(3):494-498.)
[20]李艺璇.END样本最近邻密度估计的相合性[J].湖北大学学报(自然科学版),2016,38(5):396-402.(LI Yixuan.Consistency of Nearest Neighbor Estimator of Density Function for Extended Negatively Dependent Samples[J].Journal of Hubei University(Natural Science),2016,38(5):396-402.)