多元拟合优度检验与复发事件数据统计分析
|
摘要
拟合优度检验和参数估计是统计推断中永恒的课题,结合具体模型有丰富的内容.本文主要研究了三个问题:首先,对Pearsonχ~2检验的改进,克服Pearsonχ~2检验不稳健和分组不唯一的弱点.其次,考虑了垂直密度表示的应用,提出并讨论了中心相似多元分布统计模型,并考虑了垂直密度表示在多元拟合优度检验中的应用.最后,针对复发事件数据,提出了几类回归模型,讨论了模型参数的估计及其统计性质.
在第2章中,对Pearsonχ~2检验进行了改进.一方面,提出了极大χ~2检验,给出了极大χ~2统计量的构造,并将极大χ~2检验应用于检验方向数据是否来自球面上的均匀分布,与包括Rayleigh,Ajne,Bingham,Giné检验在内的已有均匀检验进行模拟功效的比较,大量的模拟结果表明:针对不同的对立假设,极大χ~2检验有较高的功效且是稳健的.另一方面,对Pearsonχ~2检验的分组原则进行了改进,提出了按概率密度函数值(纵坐标)进行分组的原则,给出了分组点具体的计算公式.克服了Pearsonχ~2检验传统(横坐标)分组不唯一的弱点.大量的模拟研究表明:相比较传统的分组原则,按概率密度函数值分组具有更高的模拟功效.
在第3章中,基于垂直密度表示理论,提出了中心相似多元分布统计模型,多元正态分布是其特例.该模型为多兀分布的密度构造提供了一种可行的方法.首先,对所提出的模型中的未知参数,利用矩估计的方法,给出了未知参数矩估计一般的计算表达式.并证明了所得矩估计的渐近性质,通过一些例子说明了该模型的应用.其次,利用极大似然的思想,考虑了中心相似多元分布统计模型中未知参数的极大似然估计,同时给出了一般的估计方程组.最后,讨论了垂直密度表示在多元拟合优度检验中的应用,包括球对称分布的拟合优度检验,χ~2检验的VDR分组,中心相似分布的拟合优度检验.
在第4章上,在复发事件数据下,提出了几类回归模型.首先,对单类型复发事件数据,提出了加性乘积比率回归模型,该模型包含了一大类比率回归模型,加性比率回归模型和乘性比率回归模型是其特殊情形.利用估计方程的思想,给出了该模型中未知参数和非参数函数的一种估计方法.利用现代经验过程理论,证明了所得估计的渐近性质.并将该模型应用于分析CGD数据.其次,对多类型复发事件数据,同样提出了加性乘积比率回归模型,讨论了该模型中未知参数的一种估计方法,在一些正则条件下,证明了所得估计的相合性和渐近正态性.最后,在多类型复发事件数据下,提出了变系数加性乘积比率回归模型,讨论了所提出模型未知参数的估计及其大样本性质.
Goodness-of fit and parameter estimation are the eternal topic in statistical inference.They have various contents in different models.In this thesis,three problems are studied.First,two approaches are proposed in order to modify Pearson's chi-squared test.These modified tests remove the weakness that Pearson's chi-squared test is not stable and partition of sample space is not unique. Second,some applications of vertical density representation in goodness of fit tests of multivariate distribution are considered.Finally,several regression models for recurrent event data are proposed.And the unknown parameters in these models are estimated.The resulting estimators are proven to be consistent and asymp-totically normal.
In Chapter 2,two methods are proposed to modify Pearson's chi-squared test. On the one hand,maximized chi-squared test is proposed.A construction of the maximized chi-squared test statistic is obtained.And the maximized chi-squared test is applied to test whether the vectorial data come from the uniformity defined on the hypersphere.Tests include the maximized chi-squared test,Rayleigh,Ajne, Giné,and Bingham tests,are compared the empirical power against the hypothesis of a Von Mises-Fisher distribution or a Watson distribution in some cases. The simulation results show that the maximized chi-squared test is stable against different alternative.On the other hand,based on the value of probability density function,a new principle of partition of classes is proposed.Furthermore,a formula to calculate division points is represented.The new principle removes the weakness that the traditional partition of classes is not unique for the same sample. The simulation studies demonstrate that Pearson's chi-squared test based on new principle is more powerful than that based on traditional partition in abscissa.
In Chapter 3,based on the results of vertical density representation and center-similar distribution,a statistical model of center-similar multivariate distribution is proposed.The proposed model includes multivariate normal distribution as a special case.Firstly,the unknown parameters of the proposed model are estimated by method of moment.The asymptotic properties of the resulting estimators are established.Some examples are presented to illustrate the application of the proposed model.Secondly,by maximum likelihood method,the estimators of unknown parameters in the proposed model are obtained.The system of estimation equations is presented.Finally,results of vertical density representation are used to goodness-of-fit tests of multivariate distribution,including goodness-of-fit tests of spherically symmetric distribution and center-similar distribution, partition ofχ~2 test through vertical density representation.
In Chapter 4,for recurrent event data,several regression models are proposed. Firstly,a class of general additive-multiplicative rates models for single type recurrent event is proposed.The proposed models include the additive rates and multiplicative rates models as special cases.For the inference on the model parameters,estimating equation approaches are developed,and asymptotic properties of the proposed estimators are established through modern empirical process theory.In addition,the proposed models are applied to multiple-infection data from a clinic study on chronic granulomatous disease(CGD).Secondly,general additive-multiplicative rates models for multiple type recurrent event data also are considered.We formulate estimating equations for unknown parameters and nonparametric function of proposed models.Under some regularity conditions,the consistency and asymptotic normality of resulting estimators are shown.Finally, we present a flexible additive-multiplicative rates model for multiple type recurrent event data.Procedures for making inference about the model parameters are provided.Asymptotic properties of the proposed estimators are established.
在第2章中,对Pearsonχ~2检验进行了改进.一方面,提出了极大χ~2检验,给出了极大χ~2统计量的构造,并将极大χ~2检验应用于检验方向数据是否来自球面上的均匀分布,与包括Rayleigh,Ajne,Bingham,Giné检验在内的已有均匀检验进行模拟功效的比较,大量的模拟结果表明:针对不同的对立假设,极大χ~2检验有较高的功效且是稳健的.另一方面,对Pearsonχ~2检验的分组原则进行了改进,提出了按概率密度函数值(纵坐标)进行分组的原则,给出了分组点具体的计算公式.克服了Pearsonχ~2检验传统(横坐标)分组不唯一的弱点.大量的模拟研究表明:相比较传统的分组原则,按概率密度函数值分组具有更高的模拟功效.
在第3章中,基于垂直密度表示理论,提出了中心相似多元分布统计模型,多元正态分布是其特例.该模型为多兀分布的密度构造提供了一种可行的方法.首先,对所提出的模型中的未知参数,利用矩估计的方法,给出了未知参数矩估计一般的计算表达式.并证明了所得矩估计的渐近性质,通过一些例子说明了该模型的应用.其次,利用极大似然的思想,考虑了中心相似多元分布统计模型中未知参数的极大似然估计,同时给出了一般的估计方程组.最后,讨论了垂直密度表示在多元拟合优度检验中的应用,包括球对称分布的拟合优度检验,χ~2检验的VDR分组,中心相似分布的拟合优度检验.
在第4章上,在复发事件数据下,提出了几类回归模型.首先,对单类型复发事件数据,提出了加性乘积比率回归模型,该模型包含了一大类比率回归模型,加性比率回归模型和乘性比率回归模型是其特殊情形.利用估计方程的思想,给出了该模型中未知参数和非参数函数的一种估计方法.利用现代经验过程理论,证明了所得估计的渐近性质.并将该模型应用于分析CGD数据.其次,对多类型复发事件数据,同样提出了加性乘积比率回归模型,讨论了该模型中未知参数的一种估计方法,在一些正则条件下,证明了所得估计的相合性和渐近正态性.最后,在多类型复发事件数据下,提出了变系数加性乘积比率回归模型,讨论了所提出模型未知参数的估计及其大样本性质.
Goodness-of fit and parameter estimation are the eternal topic in statistical inference.They have various contents in different models.In this thesis,three problems are studied.First,two approaches are proposed in order to modify Pearson's chi-squared test.These modified tests remove the weakness that Pearson's chi-squared test is not stable and partition of sample space is not unique. Second,some applications of vertical density representation in goodness of fit tests of multivariate distribution are considered.Finally,several regression models for recurrent event data are proposed.And the unknown parameters in these models are estimated.The resulting estimators are proven to be consistent and asymp-totically normal.
In Chapter 2,two methods are proposed to modify Pearson's chi-squared test. On the one hand,maximized chi-squared test is proposed.A construction of the maximized chi-squared test statistic is obtained.And the maximized chi-squared test is applied to test whether the vectorial data come from the uniformity defined on the hypersphere.Tests include the maximized chi-squared test,Rayleigh,Ajne, Giné,and Bingham tests,are compared the empirical power against the hypothesis of a Von Mises-Fisher distribution or a Watson distribution in some cases. The simulation results show that the maximized chi-squared test is stable against different alternative.On the other hand,based on the value of probability density function,a new principle of partition of classes is proposed.Furthermore,a formula to calculate division points is represented.The new principle removes the weakness that the traditional partition of classes is not unique for the same sample. The simulation studies demonstrate that Pearson's chi-squared test based on new principle is more powerful than that based on traditional partition in abscissa.
In Chapter 3,based on the results of vertical density representation and center-similar distribution,a statistical model of center-similar multivariate distribution is proposed.The proposed model includes multivariate normal distribution as a special case.Firstly,the unknown parameters of the proposed model are estimated by method of moment.The asymptotic properties of the resulting estimators are established.Some examples are presented to illustrate the application of the proposed model.Secondly,by maximum likelihood method,the estimators of unknown parameters in the proposed model are obtained.The system of estimation equations is presented.Finally,results of vertical density representation are used to goodness-of-fit tests of multivariate distribution,including goodness-of-fit tests of spherically symmetric distribution and center-similar distribution, partition ofχ~2 test through vertical density representation.
In Chapter 4,for recurrent event data,several regression models are proposed. Firstly,a class of general additive-multiplicative rates models for single type recurrent event is proposed.The proposed models include the additive rates and multiplicative rates models as special cases.For the inference on the model parameters,estimating equation approaches are developed,and asymptotic properties of the proposed estimators are established through modern empirical process theory.In addition,the proposed models are applied to multiple-infection data from a clinic study on chronic granulomatous disease(CGD).Secondly,general additive-multiplicative rates models for multiple type recurrent event data also are considered.We formulate estimating equations for unknown parameters and nonparametric function of proposed models.Under some regularity conditions,the consistency and asymptotic normality of resulting estimators are shown.Finally, we present a flexible additive-multiplicative rates model for multiple type recurrent event data.Procedures for making inference about the model parameters are provided.Asymptotic properties of the proposed estimators are established.
引文
1 K.Pearson.On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling.Philosophical Magazine.1900,50:157-172.
2 J.Neyman,E.S.Pearson.On the use and interpretation of criteria for purposes of statistical inference I.Biometrika.1928,20A:175-240.
3 J.Neyman.Contribution to the theory of the χ~2 test.Proceedings of the Berkeley Symposium on Mathematical Statistics and Probability,Berkeley and Los Angeles,University of California Press.1949,239-273.
4 S.E.Fienberg.The use of chi-squared statistics for categorical data problems.J.Roy.Statist.Sco.B.1979,41:54-64.
5 N.Cressie,T.R.C.Read.Multinomial goodness-of-fit tests.J.Roy.Statist.Sco.B.1984,46:440-464.
6 J.Neyman.'Smooth' test for goodness of fit.Skand.Aktuarietidskr.1937,20:150-199.
7 杨振海.拟合优度检验.合肥:安徽教育出版社,1994.
8 R.A.Fisher.The conditions under which χ~2 measures the discrepancy between observation and hypothesis.J.Roy.Statist.Soc.1924,87:442-450.
9 H.Chernoff,E.L.Lehmann.The use of maximum-likelihood estimates in χ~2 test for goodness fit.Ann.Math statist.1954,25:579-586.
10 H.B.Mann,A.Wald.On the choice of the number of class intervals in the application of the Chi-square test.Ann.Math.Statist.1942,13:306-317.
11 W.G.Cochran.Some methods of strengthening the common χ~2 tests.Biometrics.1954,10:417-451.
12 J.T.Roscoe,J.A.Byars.An investigation of the restraints with respect to sample size commonly imposed on the use of the Chi-square statistic.J.Amer.Statist.Assoc.1971,66:755-759.
13 J.Oosterhoff.The choice of cells in chi-square tests.Statist.Neerl.1985,39:115-128.
14 M.P.Quine,J.Robinson.Efficiencies of chi-square and likelihood ratio goodness-of-fit tests.Ann.Statist.1985,13:727-742.
15 J.C.W.Rayner,D.J.Best,K.G.Dodds.The construction of the simple χ~2 and Neyman smooth goodness of fit tests.Statistica Neerlandica.1985,39:35-50.(or 25-50)
16 T.Inglot,W.C.M.Nallenberg,T.Ledwina.Power approximations to and power comparison of smooth goodness-of-fit tests.Scand J.Statist.1994,21:131-145.
17 Yu.I.Ingster.Adaptive chi-square tests.Zapiski Nauchnych Seminarov LOMI.1997,244:150-166.
18 M.Bogdan.Data driven versions of Pearson's chi-square test for uniformity.J.Statist.Comput.Simul.1995,52:217-237.
19 P.Hall,E.J.Hannah.On stochastic complexity and nonparametric density estimation.Biometrih.1988,75:705-714.
20 J.Rissanen T.P.Speed,B.Yu.Density estimation by stochastic complexity.IEEE Trans.Inform.Theory.1992,38:315-323.
21 O.Thas.Nonparametrical tests based on sample space partitions.Ph.D.Thesis.Universiteit Gent.2001.
22 W.C.M.Kallenberg,J.Oosterhoff,B.F.Schriever.The number of the classes in chi-squared goodness of fit tests.J.Amer.Statist.Assoc.1985,80:959-968.
23 T.Inglot,A.Janic-Wroblewska.Data driven chi-square test for uniformity with unequal cells.J.Statist.Comput.Simul.2003,73(8):545-561.
24 M.D.Troutt.Vertical density representation and a further remark on the Box-Muller method.Statistics.1993,24:81-83.
25 M.D.Troutt.A theorem on the density of the density ordinate and an alternative derivation of the Box-Muller method.Statistics.1991,22:436-466.
26 K.T.Fang.Z.H.Yang.S.Kotz.Generation of Multivariate Distribution by Vertical Density Representation.Statistics.2001.35:281-293
27 W.K.Pang.Z.H.Yang.S.H.Hou.M.D.Troutt.Further Results on Multivariate Vertical Density Representation and an Application to Random Vector Generation.Statistics.2001.35:467-477.
28 Z.H.Yang.S.Kotz.Center-similar Distribution with Application in Multivariate Analysis.Statistics & Probability Letters.2003,l64:335-345
29 K.T.Fang.Y.Wang.Number-theoretic Methods in Statistics.London:Chapman& Hall,1994.
30杨振海,程维虎.垂直密度表示及其应用.应用概率统计,2006,22(3):329-336.
31 Zhenhai Yang.W.K.Pang.S.H.Hou,P.K.Leung.On a combination method of VDR and patchwork for generating uniform random points on a unit sphere.Journal of Multivariate Analysis.2005,95:23-36.
32 M.D.Troutt,W.K.Pang,S.H.Hou.Vertical Density Representation and its Applications.Singapore:W-orld Scientific Publishing,2004.
33 Zhenhai Yang,K.T.Fang,S.Kotz.On the Student's t-distribution and the t-statistic.Journal of Multivariate Analysis.2007,98:1293-1304.
34 J.G.Mauldon.Characterizing properties of statistical distributions.Quart.J.Math.1956,2(7):155-160.
35 R.L.Prentice,B.J.Williams,A.V.Peterson,On the regression analysis of multivariate failure time data.Biometrika.1981,68:373-379.
36 M.S.Pepe,J.Cal.Some graphical displays and marginal regression analyses for recurrent failure times and time-dependent covariates.J.Amer.Statist.Assoc.,1993,88:811-820.
37 H.Abu-Libdeh,B.W.Turnbull,L.C.Clark.Analysis of multi-type recurrent events in longitudinal studies:Application to a skin cancer prevention trial.Biometrics,1990,46:1017-1034.
38 L.J.Wei D.Y.Lin,L.Weissfeld.Regression analysis of multivariate incomplete failure time data by modeling marginal distributions.J.Amer.Statist.Assoc.,1989,84:1065-1073.
39 E.W.Lee,L.J.Wei,D.A.Amato.Cox-type regression analysis for large numbers of small groups of correlated failure time observations.In:Klein J P,Goel P K(eds)Survival analysis:state of the art.Dordrecht:Kluwer,1992.237-247.
40 P.K.Andersen,R.D Gill.Cox's regression model for counting processes:a large sample study.Ann.Statist.,1982,10:1100-1120.
41 S.H.Chang,M.C Wang.Conditional regression analysis for time data.J.Amer star.Assoc.,1999,94:1221-1230.
42 M.C.Wang,Y.Q.Chen.Nonparametric and semiparametric trend analysis for stratified recurrence time data.Biometrics,2000,56:789-794.
43 S.M.Ross.Introduction to Probability Models.New York:Academic Press.1989.
44 D.Y.Lin,L.J.Wei,I.Yang,Z.Ying.Semiparametric regression for the mean and rate functions of recurrent events.J.R.Statist.Soc.B,2000,62(4):711-730.
45 D.Y.Lin,Z.LYing,Semiparametric analysis of the additive risk model.Biometrika,1994,81(1):61-71.
46 D.E.Schaubel,D.Zeng,J.Cal.A semiparametric additive rates model fbr recurrent event data.Lifetime Data Analysis.2006,12:387-406.
47 D.R.Cox.Regression models and life-tables(with discussion).J.Roy.Statist.Soc.Ser.B.1972,34:187-220.
48 N.Breslow.Contribution to the discussion of the paper by D.R.Cox.J.Roy.Statist.Soc.Ser.B.1974,34:187-220.
49 D.Y.Lin,L.J.Wei.The robust inference for Cox proportional hazards model.J.Amer.Statist.Assoc.1989,84:1074-1078.
50 J.F.Lawless,C.Nadeau,Some simple robust methods fbr the analysis of recurrent events.Technometrics.1995,37:158-168.
51 J.F.Lawless,C.Nadeau,R.J.Cook.Analysis of mean and rate functions for recurrent events.In Proc.1st Seattle Symp.Biostatistcs:Survival analysis.New York:Springer,1997.37-49.
52 K.Y.Liang,S.G.Self,Y.C.Chang.Modelling marginal hazards in multivariate failure time data.J.Roy.Statist.Soc.Set.B.1993,55:441-453.
53 D.Y.Lin,L.J.Wei,Z.Ying.Accelerated failure time models for counting processes.Biometrika.1998,85:605-618.
54 D.Y.Lin,L.J.Wei,Z.Ying.Semiparametric transformation models for point processes.J.Amer.Statist.Assoc.2001,96:620-628.
55 A.Mailer,L.Sun,X.Zhou.Estimating the expected total number of events in a process.J.Amer.Statist.Assoc.2002,97:577-589.
56 L.Sun,B.Su.A class of accelerated means regression models for recurrent even data.Lifetime Data Analysis.2008,14:357-375.
57 J.Cai,R.L.Prentice.Regression estimation using multivariate failure time data and a common baseline hazard function model.Lifetime Data Anal.1997,3:197-213.
58 D.Zeng,D.Y.Lin.Effective estimation of semiparametric transformation models for counting processes.Biometrika.2006,93:627-640.
59 D.Zeng,D.Y.Lin.Semiparametric transformation models with random effects for recurrent events.J.Amer.Statist.Assoc.2007,102:167-180.
60 Y.Huang,Y.Q.Chen.Marginal regression of gaps between recurrent events.Lifetime Data Analysis.2003,9:293-303.
61 M.C.Wang,S.H.Chang.Nonparametric estimation of recurrent survival function.J.Amer.Statist.Assoc.1999,94:146-153.
62 M.C.Wang,M.T.Wells.Nonparametric estimation of successive duration times under dependent censoring.Biometrika.1998,85:561-572.
63 D.Y.Lin,W.Sun,Z.Ying.Nonparametric estimation of the gap time distributions for serial events with censored data.Biometrika.1999,86:59-70.
64 Y.Huang.Multistate accelerated sojourn times model.J.Amer.Statist.Assoc.2000,95:619-627.
65 D.E.Schaubel,J.Cai.Non-parametric estimation of gap time survival functions for ordered multivariate failure time data.Statist.Med.2004a,23:1885-1900.
66 D.E.Schaubel,J.Cal.Regression methods fbr gap time hazard function of sequentially ordered multivariate failure time data.Biometrika.2004b,91:291-303..
67 L.Sun,D.Park,J.Sun.The additive hazard model fbr recurrent gag times.Statist.Sinica.2006,16:919-932.
68 S.H.Chang.Estimating marginal effects in accelerated failure time models for serial sojourn times among repeated events.Lifetime Data Analysis.2004,10:175-190.
69 R.Strawderman.The accelerated gap times model.Biometrika.2005,92:647-666.
70 W.Lu.Marginal regression of multivariate event times based on linear transformation models.Lifetime Data Analysis.2005,11:389-404.
71 D.Ghosh,D.Y.Lin.Nonparametric analysis of recurrent events and death.Biometrics.2000,56:554-562.
72 D.Ghosh,D.Y.Lin.Marginal regression models for recurrent and terminal events.Statistica Sinica.2002,12:663-688.
73 D.Ghosh,D.Y.Lin.Semiparametric analysis of recurrent event data in the presence of dependent censoring.Biometrics.2003,59:877-885.
74 M.C.Wang,J.Qin,C.T.Chiang.Analyzing recurrent event data with informative censoring.J.Amer.Statist.Assoc.2001,96:1057-1065.
75 M.Miloslavsky,S.Keles,M.van der Laan.Recurrent events analysis in the presence of time-dependent covariates and dependent censoring.J.R.Statist.Soc.B.2004,66:239-257.
76 C.H.Huang,M.C.Wang.Joint modeling and estimation for recurrent event processes and failure time data.J.Amer.Statist.Assoc.2004,99:1153-1165.
77 L.Liu,R.A.Wolfe,X.Huang.Shared frailty models for recurrent events and a terminal event.Biometrics.2004,60:747-756.
78 Y.Ye,J.D.Kalbfleisch,D.E.Schaubel.Semiparametric analysis of correlated recurrent and terminal events.Biometrics.2007,63:78-87.
79 X.Huang,L.Liu.A joint frailty model for survival and gap times between recurrent events.Biometrics.2007,63:389-397.
80 M.C.Wang,C.T.Chiang.Non-parametric methods for recurrent event data with informative and non-informative censorings.Statist.Med.2002,21:445-456.
81 D.Sinha,T.Maiti.A Bayesian approach for the analysis of panel count data with dependent termination.Biometrics.2004,60:34-40.
82 C.T.Chiang,L.F.James,M.C.Wang.Random weighted bootstrap method for recurrent events with informative censoring.Lifetime Data Analysis.2005,11:489-509.
83 J.W.Cai,D.E.Schaubel.Marginal means/rates models for multiple type recurrent event data.Lifetime Data Analysis.2004,10:121-138.
84 D.E.Schaubel,J.W.Cal.Rate/mean regression for multiple-sequence recurrent event data with missing event category.Scandinavian Journal of Statistics.2006a,33:191-207.
85 D.E.Schaubel,J.W.Cal.Multiple imputation methods for recurrent event data with missing event category.Canadian Journal of Statistics.2006b,34:677-692.
86 B.E.Chen,R.J.Cook.Tests for multivariate recurrent events in the presence of a terminal event:application to studies of cancer metastatic to bone.Biostatistics.2004,5:129-143.
87 R.J.Cook,J.F.Lawless.The Statistical Analysis of Recurrent Events.Springer,New York,2007.
88 Luo Xianhua,Jiang Fangjiao,Sun Liuquan,Yang Zhenhai.Marginal regression of multiple type recurrent event data based on transformation models.Chinese Journal of Engineering Mathematics.2008,25(2):326-332.
89 W.G.Cochran.The χ~2 test of goodness of fit.Annals of Mathematical Statistics.1952.23:315-345.
90 R.L.Plackett.The Analysis of Categorical Data(2nd edition).London:Griffin,1981.
91 A.Cohen,H.B.Sackrowitz.Unbiasedness of the chi-square,likelihood ratio,and other goodness of fit tests for the equal cell case.Annals of Statistics.1975,3:959-964.
92 K.Larntz.Small-sample comparisons of exact levels for chi-squared goodness-of-fit statistics.J.Amer.Statist.Assoc.1978,73:253-263.
93 H.B.Lawal.Tables of percentage points of Pearson's goodness-of-fit statistic for use with small expectations.Applied Statistics.1980,29:292-298.
94 K.J.Koehler,K.Larntz.An empirical investigation of goodness-of-fit statistics fbr sparse multinomials.J.Amer.Statist.Assoc.1980,75:336-344.
95 O.Nempthorne.The classical problem of inference goodness of fit.Proc.Fifth Berkeley Symp.Math.Statist.Prob.1967,1:235-249.
96 M.Sibuya.A method fbr generating unifbrmly distributed points on ndimensional spheres.Ann.Inst.Statist.Math.,1962,14:81-85.
97 P.J.Diggle,N.I.FISHER,A.J.LEE.A comparison of tests of unifbrmity for spherical data.Austral.J.Statist.1985,27(1):53-59.
98 R.J.Beran.Testing for unifbrmity on a compact homogeneous space.J.Appl.Probab.1968,5:177-195.
99 E.Gin(?).Invariant tests of unifbrmity on compact Riemannian manifblds based on Sobolev norms.Ann.Statist.1975,386:1243-1266.
100 A.Figueiredo,P.Gomes.Power of tests of uniformity defined on the hypersphere.Comm.Statist.Simulation Comput.2003,32(1):87-94.
101 A.Figueiredo.Comparison of tests of uniformity defined on the hypersphere.Statist.Probab.Lett.2007,77(3):329-334.
102 A.Wood.Simulation of the Von Mises-Fisher distribution.Comm.Statist.Simulation Comput.1994,23(1):157-164.
103 Robb J.Muirhead.Aspects of Multivariate Statistical Theory.New York:John Wiley & Sons,1982.
104 "Student".On the probable error of the mean.Biometrih.1908,6:1-25.
105 A.M.Kagan,Y.V.Linnik,C.R.Rao.Characterization problems in Mathematical Statistics.New York:Wiley,1973.
106 B.Efron.Student's t-test under symmetry conditions.J.Amer.Statist.Asoc.1969,64:1278-1302.
107 L.Bondesson.Characterizations of probability laws through constant regression.Z.Wahrscheinlichkeitstheorie verw.Gebiete.1974,30:93-115.
108 L.Bondesson.When is the t-statistic t-distributed.Sankhy(?),Ser.A.1983,45:338-345.
109 K.T.Fang,S.Kotz,K.W.Ng.Symmetric multivariate and related distributions.London:Chapman and Hall,1990.
110 D.Y.Lin,Z.L.Ying.Semiparametric analysis of general additivemultiplicative hazard models for counting processes.Ann.Statist.1995,23(5):1712-1734.
111 D.P.Byar.The Veterans Administration Study of chemoprophylaxis for recurrent stage Ⅰ bladder tumors:comparisons of placebo,pyridoxine,and topical thitepa,in bladder tumors and other topics in Urological oncology,edited by M.Pavone-Macaluso,P.H.Smith,and F.Edsmyn,New York:Plenum Press.1980,363-370.
112 Y.Bilias,M.Gu,Z.Ying.Towards a general asymptotic theory for Cox model with staggered entry.Ann.Statist.,1997,25:662-682.
113 D.Pollard.Empirical Processes:Theory and Applications.Hayward:Institute of Mathematical Statistics,1990.
114 Ⅰ.Karatzas,S.E.Shreve.Brownian motion and stochastic calculus.New York:Springer,1988.
115 G.R.Shorack,J.A.Wellner.Empirical processes with applications to statistics.New York:Wiley,1986.
116 C.Goffman.Calculus of Several Variables.New York:Harper and Row,1965.
117 T.R.Fleming,D.P.Harrington.Counting processes and survival analysis.New York:Wiley,1991.
118 D.Schaubel,H.Johansen,M.Durra,M.Desmeules,A.Backer,Y.Mao.Neonatal characteristics as risk factors for preschool asthma.J.Asthma,1996,33:255- 264
119 O.O.Aalen.A model for non-parametric regresssion analysis of counting processes.In Lecture Notes in Statistics-2:Mathematical Statistics and Probability Theory.New York:Springer-Verlag.1980,1-25.
120 G.N.Hortobagyi,R.Theriault,A.Lipton,L.Porter,D.Blayney,C,Sinoff,H.Wheeler,J.F.Simeone,J.Seaman,R.D.Knight,M.Heffernan,K.Mellars,D.J.Reitsma.Long-term prevention of skeletal complications of metastatic breast cancer with pamidronate.Journal of Clinical Oncology.1998,16:2038-2044.
121 T.Martinussen,T.H.Scheike.A flexible additive multiplicative hazard model.Biometrika.2002,89(2):283-298.
122戴家佳,何穗.多类型复发事件下的加性乘积比率回归模型.工程数学学报.2008,25(6):979-988.
123刘焕彬,何穗,孙六全.复发事件下一般半参数比率回归模型.应用数学学报.2008,31(4):671-681.
124 E.C.MacRae.Matrix devivatives with an application to an adaptive linear dicision problem[J].Ann.Statist.1974,2:337-346.
125 P.K.Andersen,Borgan φ,R.D Gill,N.Keiding.Statistical models based on counting process.New York:Springer-Verlag,1993
126 D.R.Cox,D.Oakes.Analysis of Survival Data.London:Chapman and Hall,1984.
127苏岩.拟合优度检验与统计模型有效性研究.北京工业大学博士论文,2007.
128 G.R.Shorack.Probability for Statisticians.New York:Springer-Verlag,2000.
129 J.F.Lawless.Statistical Models and Methods for Lifetime Data(second Edition).New York:John Wiley & Sons,2002.
130 E.T.Lee,J.W.Wang.Statistical Methods for Survival Data Analysis(Third Edition).New York:John Wiley & Sons,2003.
131 J.D.Kalbfleisch,R.L.Prentice.The Statistical Analysis of Failure Time Data.New York:John Wiley & Sons,2002.
132方开泰,张尧庭.广义多元分析.科学出版社,北京,1993.
2 J.Neyman,E.S.Pearson.On the use and interpretation of criteria for purposes of statistical inference I.Biometrika.1928,20A:175-240.
3 J.Neyman.Contribution to the theory of the χ~2 test.Proceedings of the Berkeley Symposium on Mathematical Statistics and Probability,Berkeley and Los Angeles,University of California Press.1949,239-273.
4 S.E.Fienberg.The use of chi-squared statistics for categorical data problems.J.Roy.Statist.Sco.B.1979,41:54-64.
5 N.Cressie,T.R.C.Read.Multinomial goodness-of-fit tests.J.Roy.Statist.Sco.B.1984,46:440-464.
6 J.Neyman.'Smooth' test for goodness of fit.Skand.Aktuarietidskr.1937,20:150-199.
7 杨振海.拟合优度检验.合肥:安徽教育出版社,1994.
8 R.A.Fisher.The conditions under which χ~2 measures the discrepancy between observation and hypothesis.J.Roy.Statist.Soc.1924,87:442-450.
9 H.Chernoff,E.L.Lehmann.The use of maximum-likelihood estimates in χ~2 test for goodness fit.Ann.Math statist.1954,25:579-586.
10 H.B.Mann,A.Wald.On the choice of the number of class intervals in the application of the Chi-square test.Ann.Math.Statist.1942,13:306-317.
11 W.G.Cochran.Some methods of strengthening the common χ~2 tests.Biometrics.1954,10:417-451.
12 J.T.Roscoe,J.A.Byars.An investigation of the restraints with respect to sample size commonly imposed on the use of the Chi-square statistic.J.Amer.Statist.Assoc.1971,66:755-759.
13 J.Oosterhoff.The choice of cells in chi-square tests.Statist.Neerl.1985,39:115-128.
14 M.P.Quine,J.Robinson.Efficiencies of chi-square and likelihood ratio goodness-of-fit tests.Ann.Statist.1985,13:727-742.
15 J.C.W.Rayner,D.J.Best,K.G.Dodds.The construction of the simple χ~2 and Neyman smooth goodness of fit tests.Statistica Neerlandica.1985,39:35-50.(or 25-50)
16 T.Inglot,W.C.M.Nallenberg,T.Ledwina.Power approximations to and power comparison of smooth goodness-of-fit tests.Scand J.Statist.1994,21:131-145.
17 Yu.I.Ingster.Adaptive chi-square tests.Zapiski Nauchnych Seminarov LOMI.1997,244:150-166.
18 M.Bogdan.Data driven versions of Pearson's chi-square test for uniformity.J.Statist.Comput.Simul.1995,52:217-237.
19 P.Hall,E.J.Hannah.On stochastic complexity and nonparametric density estimation.Biometrih.1988,75:705-714.
20 J.Rissanen T.P.Speed,B.Yu.Density estimation by stochastic complexity.IEEE Trans.Inform.Theory.1992,38:315-323.
21 O.Thas.Nonparametrical tests based on sample space partitions.Ph.D.Thesis.Universiteit Gent.2001.
22 W.C.M.Kallenberg,J.Oosterhoff,B.F.Schriever.The number of the classes in chi-squared goodness of fit tests.J.Amer.Statist.Assoc.1985,80:959-968.
23 T.Inglot,A.Janic-Wroblewska.Data driven chi-square test for uniformity with unequal cells.J.Statist.Comput.Simul.2003,73(8):545-561.
24 M.D.Troutt.Vertical density representation and a further remark on the Box-Muller method.Statistics.1993,24:81-83.
25 M.D.Troutt.A theorem on the density of the density ordinate and an alternative derivation of the Box-Muller method.Statistics.1991,22:436-466.
26 K.T.Fang.Z.H.Yang.S.Kotz.Generation of Multivariate Distribution by Vertical Density Representation.Statistics.2001.35:281-293
27 W.K.Pang.Z.H.Yang.S.H.Hou.M.D.Troutt.Further Results on Multivariate Vertical Density Representation and an Application to Random Vector Generation.Statistics.2001.35:467-477.
28 Z.H.Yang.S.Kotz.Center-similar Distribution with Application in Multivariate Analysis.Statistics & Probability Letters.2003,l64:335-345
29 K.T.Fang.Y.Wang.Number-theoretic Methods in Statistics.London:Chapman& Hall,1994.
30杨振海,程维虎.垂直密度表示及其应用.应用概率统计,2006,22(3):329-336.
31 Zhenhai Yang.W.K.Pang.S.H.Hou,P.K.Leung.On a combination method of VDR and patchwork for generating uniform random points on a unit sphere.Journal of Multivariate Analysis.2005,95:23-36.
32 M.D.Troutt,W.K.Pang,S.H.Hou.Vertical Density Representation and its Applications.Singapore:W-orld Scientific Publishing,2004.
33 Zhenhai Yang,K.T.Fang,S.Kotz.On the Student's t-distribution and the t-statistic.Journal of Multivariate Analysis.2007,98:1293-1304.
34 J.G.Mauldon.Characterizing properties of statistical distributions.Quart.J.Math.1956,2(7):155-160.
35 R.L.Prentice,B.J.Williams,A.V.Peterson,On the regression analysis of multivariate failure time data.Biometrika.1981,68:373-379.
36 M.S.Pepe,J.Cal.Some graphical displays and marginal regression analyses for recurrent failure times and time-dependent covariates.J.Amer.Statist.Assoc.,1993,88:811-820.
37 H.Abu-Libdeh,B.W.Turnbull,L.C.Clark.Analysis of multi-type recurrent events in longitudinal studies:Application to a skin cancer prevention trial.Biometrics,1990,46:1017-1034.
38 L.J.Wei D.Y.Lin,L.Weissfeld.Regression analysis of multivariate incomplete failure time data by modeling marginal distributions.J.Amer.Statist.Assoc.,1989,84:1065-1073.
39 E.W.Lee,L.J.Wei,D.A.Amato.Cox-type regression analysis for large numbers of small groups of correlated failure time observations.In:Klein J P,Goel P K(eds)Survival analysis:state of the art.Dordrecht:Kluwer,1992.237-247.
40 P.K.Andersen,R.D Gill.Cox's regression model for counting processes:a large sample study.Ann.Statist.,1982,10:1100-1120.
41 S.H.Chang,M.C Wang.Conditional regression analysis for time data.J.Amer star.Assoc.,1999,94:1221-1230.
42 M.C.Wang,Y.Q.Chen.Nonparametric and semiparametric trend analysis for stratified recurrence time data.Biometrics,2000,56:789-794.
43 S.M.Ross.Introduction to Probability Models.New York:Academic Press.1989.
44 D.Y.Lin,L.J.Wei,I.Yang,Z.Ying.Semiparametric regression for the mean and rate functions of recurrent events.J.R.Statist.Soc.B,2000,62(4):711-730.
45 D.Y.Lin,Z.LYing,Semiparametric analysis of the additive risk model.Biometrika,1994,81(1):61-71.
46 D.E.Schaubel,D.Zeng,J.Cal.A semiparametric additive rates model fbr recurrent event data.Lifetime Data Analysis.2006,12:387-406.
47 D.R.Cox.Regression models and life-tables(with discussion).J.Roy.Statist.Soc.Ser.B.1972,34:187-220.
48 N.Breslow.Contribution to the discussion of the paper by D.R.Cox.J.Roy.Statist.Soc.Ser.B.1974,34:187-220.
49 D.Y.Lin,L.J.Wei.The robust inference for Cox proportional hazards model.J.Amer.Statist.Assoc.1989,84:1074-1078.
50 J.F.Lawless,C.Nadeau,Some simple robust methods fbr the analysis of recurrent events.Technometrics.1995,37:158-168.
51 J.F.Lawless,C.Nadeau,R.J.Cook.Analysis of mean and rate functions for recurrent events.In Proc.1st Seattle Symp.Biostatistcs:Survival analysis.New York:Springer,1997.37-49.
52 K.Y.Liang,S.G.Self,Y.C.Chang.Modelling marginal hazards in multivariate failure time data.J.Roy.Statist.Soc.Set.B.1993,55:441-453.
53 D.Y.Lin,L.J.Wei,Z.Ying.Accelerated failure time models for counting processes.Biometrika.1998,85:605-618.
54 D.Y.Lin,L.J.Wei,Z.Ying.Semiparametric transformation models for point processes.J.Amer.Statist.Assoc.2001,96:620-628.
55 A.Mailer,L.Sun,X.Zhou.Estimating the expected total number of events in a process.J.Amer.Statist.Assoc.2002,97:577-589.
56 L.Sun,B.Su.A class of accelerated means regression models for recurrent even data.Lifetime Data Analysis.2008,14:357-375.
57 J.Cai,R.L.Prentice.Regression estimation using multivariate failure time data and a common baseline hazard function model.Lifetime Data Anal.1997,3:197-213.
58 D.Zeng,D.Y.Lin.Effective estimation of semiparametric transformation models for counting processes.Biometrika.2006,93:627-640.
59 D.Zeng,D.Y.Lin.Semiparametric transformation models with random effects for recurrent events.J.Amer.Statist.Assoc.2007,102:167-180.
60 Y.Huang,Y.Q.Chen.Marginal regression of gaps between recurrent events.Lifetime Data Analysis.2003,9:293-303.
61 M.C.Wang,S.H.Chang.Nonparametric estimation of recurrent survival function.J.Amer.Statist.Assoc.1999,94:146-153.
62 M.C.Wang,M.T.Wells.Nonparametric estimation of successive duration times under dependent censoring.Biometrika.1998,85:561-572.
63 D.Y.Lin,W.Sun,Z.Ying.Nonparametric estimation of the gap time distributions for serial events with censored data.Biometrika.1999,86:59-70.
64 Y.Huang.Multistate accelerated sojourn times model.J.Amer.Statist.Assoc.2000,95:619-627.
65 D.E.Schaubel,J.Cai.Non-parametric estimation of gap time survival functions for ordered multivariate failure time data.Statist.Med.2004a,23:1885-1900.
66 D.E.Schaubel,J.Cal.Regression methods fbr gap time hazard function of sequentially ordered multivariate failure time data.Biometrika.2004b,91:291-303..
67 L.Sun,D.Park,J.Sun.The additive hazard model fbr recurrent gag times.Statist.Sinica.2006,16:919-932.
68 S.H.Chang.Estimating marginal effects in accelerated failure time models for serial sojourn times among repeated events.Lifetime Data Analysis.2004,10:175-190.
69 R.Strawderman.The accelerated gap times model.Biometrika.2005,92:647-666.
70 W.Lu.Marginal regression of multivariate event times based on linear transformation models.Lifetime Data Analysis.2005,11:389-404.
71 D.Ghosh,D.Y.Lin.Nonparametric analysis of recurrent events and death.Biometrics.2000,56:554-562.
72 D.Ghosh,D.Y.Lin.Marginal regression models for recurrent and terminal events.Statistica Sinica.2002,12:663-688.
73 D.Ghosh,D.Y.Lin.Semiparametric analysis of recurrent event data in the presence of dependent censoring.Biometrics.2003,59:877-885.
74 M.C.Wang,J.Qin,C.T.Chiang.Analyzing recurrent event data with informative censoring.J.Amer.Statist.Assoc.2001,96:1057-1065.
75 M.Miloslavsky,S.Keles,M.van der Laan.Recurrent events analysis in the presence of time-dependent covariates and dependent censoring.J.R.Statist.Soc.B.2004,66:239-257.
76 C.H.Huang,M.C.Wang.Joint modeling and estimation for recurrent event processes and failure time data.J.Amer.Statist.Assoc.2004,99:1153-1165.
77 L.Liu,R.A.Wolfe,X.Huang.Shared frailty models for recurrent events and a terminal event.Biometrics.2004,60:747-756.
78 Y.Ye,J.D.Kalbfleisch,D.E.Schaubel.Semiparametric analysis of correlated recurrent and terminal events.Biometrics.2007,63:78-87.
79 X.Huang,L.Liu.A joint frailty model for survival and gap times between recurrent events.Biometrics.2007,63:389-397.
80 M.C.Wang,C.T.Chiang.Non-parametric methods for recurrent event data with informative and non-informative censorings.Statist.Med.2002,21:445-456.
81 D.Sinha,T.Maiti.A Bayesian approach for the analysis of panel count data with dependent termination.Biometrics.2004,60:34-40.
82 C.T.Chiang,L.F.James,M.C.Wang.Random weighted bootstrap method for recurrent events with informative censoring.Lifetime Data Analysis.2005,11:489-509.
83 J.W.Cai,D.E.Schaubel.Marginal means/rates models for multiple type recurrent event data.Lifetime Data Analysis.2004,10:121-138.
84 D.E.Schaubel,J.W.Cal.Rate/mean regression for multiple-sequence recurrent event data with missing event category.Scandinavian Journal of Statistics.2006a,33:191-207.
85 D.E.Schaubel,J.W.Cal.Multiple imputation methods for recurrent event data with missing event category.Canadian Journal of Statistics.2006b,34:677-692.
86 B.E.Chen,R.J.Cook.Tests for multivariate recurrent events in the presence of a terminal event:application to studies of cancer metastatic to bone.Biostatistics.2004,5:129-143.
87 R.J.Cook,J.F.Lawless.The Statistical Analysis of Recurrent Events.Springer,New York,2007.
88 Luo Xianhua,Jiang Fangjiao,Sun Liuquan,Yang Zhenhai.Marginal regression of multiple type recurrent event data based on transformation models.Chinese Journal of Engineering Mathematics.2008,25(2):326-332.
89 W.G.Cochran.The χ~2 test of goodness of fit.Annals of Mathematical Statistics.1952.23:315-345.
90 R.L.Plackett.The Analysis of Categorical Data(2nd edition).London:Griffin,1981.
91 A.Cohen,H.B.Sackrowitz.Unbiasedness of the chi-square,likelihood ratio,and other goodness of fit tests for the equal cell case.Annals of Statistics.1975,3:959-964.
92 K.Larntz.Small-sample comparisons of exact levels for chi-squared goodness-of-fit statistics.J.Amer.Statist.Assoc.1978,73:253-263.
93 H.B.Lawal.Tables of percentage points of Pearson's goodness-of-fit statistic for use with small expectations.Applied Statistics.1980,29:292-298.
94 K.J.Koehler,K.Larntz.An empirical investigation of goodness-of-fit statistics fbr sparse multinomials.J.Amer.Statist.Assoc.1980,75:336-344.
95 O.Nempthorne.The classical problem of inference goodness of fit.Proc.Fifth Berkeley Symp.Math.Statist.Prob.1967,1:235-249.
96 M.Sibuya.A method fbr generating unifbrmly distributed points on ndimensional spheres.Ann.Inst.Statist.Math.,1962,14:81-85.
97 P.J.Diggle,N.I.FISHER,A.J.LEE.A comparison of tests of unifbrmity for spherical data.Austral.J.Statist.1985,27(1):53-59.
98 R.J.Beran.Testing for unifbrmity on a compact homogeneous space.J.Appl.Probab.1968,5:177-195.
99 E.Gin(?).Invariant tests of unifbrmity on compact Riemannian manifblds based on Sobolev norms.Ann.Statist.1975,386:1243-1266.
100 A.Figueiredo,P.Gomes.Power of tests of uniformity defined on the hypersphere.Comm.Statist.Simulation Comput.2003,32(1):87-94.
101 A.Figueiredo.Comparison of tests of uniformity defined on the hypersphere.Statist.Probab.Lett.2007,77(3):329-334.
102 A.Wood.Simulation of the Von Mises-Fisher distribution.Comm.Statist.Simulation Comput.1994,23(1):157-164.
103 Robb J.Muirhead.Aspects of Multivariate Statistical Theory.New York:John Wiley & Sons,1982.
104 "Student".On the probable error of the mean.Biometrih.1908,6:1-25.
105 A.M.Kagan,Y.V.Linnik,C.R.Rao.Characterization problems in Mathematical Statistics.New York:Wiley,1973.
106 B.Efron.Student's t-test under symmetry conditions.J.Amer.Statist.Asoc.1969,64:1278-1302.
107 L.Bondesson.Characterizations of probability laws through constant regression.Z.Wahrscheinlichkeitstheorie verw.Gebiete.1974,30:93-115.
108 L.Bondesson.When is the t-statistic t-distributed.Sankhy(?),Ser.A.1983,45:338-345.
109 K.T.Fang,S.Kotz,K.W.Ng.Symmetric multivariate and related distributions.London:Chapman and Hall,1990.
110 D.Y.Lin,Z.L.Ying.Semiparametric analysis of general additivemultiplicative hazard models for counting processes.Ann.Statist.1995,23(5):1712-1734.
111 D.P.Byar.The Veterans Administration Study of chemoprophylaxis for recurrent stage Ⅰ bladder tumors:comparisons of placebo,pyridoxine,and topical thitepa,in bladder tumors and other topics in Urological oncology,edited by M.Pavone-Macaluso,P.H.Smith,and F.Edsmyn,New York:Plenum Press.1980,363-370.
112 Y.Bilias,M.Gu,Z.Ying.Towards a general asymptotic theory for Cox model with staggered entry.Ann.Statist.,1997,25:662-682.
113 D.Pollard.Empirical Processes:Theory and Applications.Hayward:Institute of Mathematical Statistics,1990.
114 Ⅰ.Karatzas,S.E.Shreve.Brownian motion and stochastic calculus.New York:Springer,1988.
115 G.R.Shorack,J.A.Wellner.Empirical processes with applications to statistics.New York:Wiley,1986.
116 C.Goffman.Calculus of Several Variables.New York:Harper and Row,1965.
117 T.R.Fleming,D.P.Harrington.Counting processes and survival analysis.New York:Wiley,1991.
118 D.Schaubel,H.Johansen,M.Durra,M.Desmeules,A.Backer,Y.Mao.Neonatal characteristics as risk factors for preschool asthma.J.Asthma,1996,33:255- 264
119 O.O.Aalen.A model for non-parametric regresssion analysis of counting processes.In Lecture Notes in Statistics-2:Mathematical Statistics and Probability Theory.New York:Springer-Verlag.1980,1-25.
120 G.N.Hortobagyi,R.Theriault,A.Lipton,L.Porter,D.Blayney,C,Sinoff,H.Wheeler,J.F.Simeone,J.Seaman,R.D.Knight,M.Heffernan,K.Mellars,D.J.Reitsma.Long-term prevention of skeletal complications of metastatic breast cancer with pamidronate.Journal of Clinical Oncology.1998,16:2038-2044.
121 T.Martinussen,T.H.Scheike.A flexible additive multiplicative hazard model.Biometrika.2002,89(2):283-298.
122戴家佳,何穗.多类型复发事件下的加性乘积比率回归模型.工程数学学报.2008,25(6):979-988.
123刘焕彬,何穗,孙六全.复发事件下一般半参数比率回归模型.应用数学学报.2008,31(4):671-681.
124 E.C.MacRae.Matrix devivatives with an application to an adaptive linear dicision problem[J].Ann.Statist.1974,2:337-346.
125 P.K.Andersen,Borgan φ,R.D Gill,N.Keiding.Statistical models based on counting process.New York:Springer-Verlag,1993
126 D.R.Cox,D.Oakes.Analysis of Survival Data.London:Chapman and Hall,1984.
127苏岩.拟合优度检验与统计模型有效性研究.北京工业大学博士论文,2007.
128 G.R.Shorack.Probability for Statisticians.New York:Springer-Verlag,2000.
129 J.F.Lawless.Statistical Models and Methods for Lifetime Data(second Edition).New York:John Wiley & Sons,2002.
130 E.T.Lee,J.W.Wang.Statistical Methods for Survival Data Analysis(Third Edition).New York:John Wiley & Sons,2003.
131 J.D.Kalbfleisch,R.L.Prentice.The Statistical Analysis of Failure Time Data.New York:John Wiley & Sons,2002.
132方开泰,张尧庭.广义多元分析.科学出版社,北京,1993.