Bartlett校正和校正经验似然的研究
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摘要
大样本或渐近统计方法是统计推断中常用的统计方法,这是因为估计或检验统计量的精确分布通常很难确定,我们必须利用其极限分布来作统计推断。基于一阶渐近分布的大样本方法的精确性在样本量较小时可能较差,所以统计学中一个重要的问题是如何改进这一方法。在这一论文,我们将考虑两种改进方法:Bartlett校正和校正经验似然。
我们首先对于生存分析中的Cox比例风险模型,考虑其回归系数的Wald检验的Bartlett型校正,证明了校正后的检验统计量收敛于卡方分布的速度为o(n~(-1)),并通过模拟验证了它对Wald检验在小样本时的改进;然后,我们利用校正经验似然的方法来构造Cox比例风险模型中回归系数的置信域,证明了校正经验似然统计量的分布收敛于卡方分布,通过模拟和基于一阶渐近分布的置信域进行比较,验证了前者对后者在小样本时的改进。最后,我们将校正经验似然的思想应用于其他的非参数似然,给出了校正指数型经验似然和校正指数型倾斜似然,证明了校正后的统计量收敛到极限分布χ~2的速度都为O(n~(-2)),而我们的模拟结果也表明校正后统计量的置信域的覆盖率与校正前相比有较大的提高。
Large sample or asymptotic methods are common methods used in statistical inference, since the exact distributions of the estimators or test statistics are usually difficult to find, and we have to use their asymptotic distributions for statistical inference.The accuracy of the large sample methods based on the first order asymptotic distribution may be low. Therefore,one important problem in statistics is how to improve these methods.In this thesis,we consider two of the methods suggested to improve them:Bartlett correction and adjusted empirical likelihood.
We first consider the Bartlett type correction to Wald test for the regression coefficient in Cox proportional hazard model.It's proved that the distribution of the Bartlett corrected test statistic converges to chi-square distribution in an order of o(n~(-1)).Monte-Carlo simulations are performed to verify the improvement of Bartlett corrected Wald test when the sample size is small.We then apply the adjusted empirical likelihood method to construct confidence regions for the regression coefficient in Cox proportional hazard model and show that the distribution of adjusted empirical likelihood statistic converges to chi-square distribution. Monte-Carlo simulations are also performed to verify the improvement of adjusted empirical likelihood confidence regions to that based on first order asymptotic distribution. Finally,we apply the idea of adjusted empirical likelihood method to other types of nonparametric likelihood.It's shown that the adjusted exponential empirical likelihood and adjusted exponential tilting statistics we suggested converge to chi-square distribution in an order of O(n~(-2)).Simulation studies we performed also show that the coverage probability of the confidence regions based on adjusted methods are more accurate than those based on unadjusted methods.
我们首先对于生存分析中的Cox比例风险模型,考虑其回归系数的Wald检验的Bartlett型校正,证明了校正后的检验统计量收敛于卡方分布的速度为o(n~(-1)),并通过模拟验证了它对Wald检验在小样本时的改进;然后,我们利用校正经验似然的方法来构造Cox比例风险模型中回归系数的置信域,证明了校正经验似然统计量的分布收敛于卡方分布,通过模拟和基于一阶渐近分布的置信域进行比较,验证了前者对后者在小样本时的改进。最后,我们将校正经验似然的思想应用于其他的非参数似然,给出了校正指数型经验似然和校正指数型倾斜似然,证明了校正后的统计量收敛到极限分布χ~2的速度都为O(n~(-2)),而我们的模拟结果也表明校正后统计量的置信域的覆盖率与校正前相比有较大的提高。
Large sample or asymptotic methods are common methods used in statistical inference, since the exact distributions of the estimators or test statistics are usually difficult to find, and we have to use their asymptotic distributions for statistical inference.The accuracy of the large sample methods based on the first order asymptotic distribution may be low. Therefore,one important problem in statistics is how to improve these methods.In this thesis,we consider two of the methods suggested to improve them:Bartlett correction and adjusted empirical likelihood.
We first consider the Bartlett type correction to Wald test for the regression coefficient in Cox proportional hazard model.It's proved that the distribution of the Bartlett corrected test statistic converges to chi-square distribution in an order of o(n~(-1)).Monte-Carlo simulations are performed to verify the improvement of Bartlett corrected Wald test when the sample size is small.We then apply the adjusted empirical likelihood method to construct confidence regions for the regression coefficient in Cox proportional hazard model and show that the distribution of adjusted empirical likelihood statistic converges to chi-square distribution. Monte-Carlo simulations are also performed to verify the improvement of adjusted empirical likelihood confidence regions to that based on first order asymptotic distribution. Finally,we apply the idea of adjusted empirical likelihood method to other types of nonparametric likelihood.It's shown that the adjusted exponential empirical likelihood and adjusted exponential tilting statistics we suggested converge to chi-square distribution in an order of O(n~(-2)).Simulation studies we performed also show that the coverage probability of the confidence regions based on adjusted methods are more accurate than those based on unadjusted methods.
引文
[1]Adimari,G.(1998).An empirical likelihood statistic for quantiles.J.Statist.Comput.Simul.60 85-95.
[2]Andersen,P.K.,Borgan,O.,Gill,R.D.and Keiding,N.(1995).Statistical Models Based on Counting Processes.Springer,New York.
[3]Andersen,P.K.and Gill,D.R.(1982).Cox's regression model for counting processes.Ann.Statist.10 1100-1120.
[4]Baggerly,K.A.(1998).Empirical likelihood as a goodness-of-fit measure.Biometrika 85535-547.
[5]Barndorff-Nielsen,O.E.and Hall,P.(1988).On the level-error after Bartlett adjustment of the likelihood ratio statistic.Biometrika 75 374-378.
[6]Bartlett,M.S.(1937).Properties of sufficiency and statistical tests.Proceedings of the Royal Society A 160 268-282.
[7]Bhattacharya,R.N.and Ghosh,J.K.(1978).On the validity of the formal edgeworth expansion.Ann.Statist.6 434-451.
[8]Box,G.E.P.(1949).A general distribution theory for a class of likelihood criteria.Biometrika 36 317-346.
[9]Chandra,T.K.(1985).Asymptotic expansions of perturbed chi-square variables.Sankhy(?) A 47 100-110.
[10]Chandra,T.K.and Mukerjee,R.(1991).Bartlett-type modification of Rao's efficient score statistic.Journal Multivariate Analysis 36 103-112.
[11]Chen,J.,Variyath,A.M.and Abraham,B.(2008).Adjusted empirical likelihood and its properties.Journal of Computational and Graphical Statistics 17 1-18.
[12]Chesher,A.and Smith,R.(1994).Bartlett corrections to likelihood ratio tests.Unpublished paper,Department of Economics,University of Bristol.
[13]Corcoran,S.A.(1998).Bartlett adjustment of empirical discrepancy statistics.Biometrika 85 967-972.
[14]Cordeiro,G.M.(1987).On the corrections to the likelihood ratio statistics.Biometrika 74265-274.
[15]Cordeiro,G.M.and Ferrari,S.L.P.(1991).A modified score test statistic having chi-squared distribution to order n~(-1).Biometrika 78 573-582.
[16]Cox,D.R.(1972).Regression models and life tables(with discussion).J.Roy.Statist.Soc.Set.B.34 187-220.
[17]Cox,D.R.(1988).Some aspects of conditional and asymptotic inference:A review.Sankhy(?)A 50 314-337.
[18]Cribari-Neto,F.and Cordeiro,G.M.(1996).On Bartlett and Bartlett-type corrections.Econometric Reviews 15 339-367.
[19]DiCiccio,T.,Hall,P.and Romano,J.(1991).Empirical likelihood is Bartlett correctable.The Ann.Statist.19 1053-1061.
[20]Efron,B.(1981).Nonparametric standard errors and confidence intervals(with discussion).Canad.J.Statist.9 139-172.
[21]Ferrari,S.L.P.and Cribari-Neto,F.(1993).On the corrections to the Wald test of non-linear restrictions.Economics Letters 42 321-326.
[22]Fleming,T.R.and Harrington,D.P.(1991).Counting Processes and Survival Analysis.Wiley,New York.
[23]Gu,M.(1992).On the Edgeworth expansion and bootstrap approximation for the Cox regression model under random censorship.The Canadian Joural of Statistics 20 399-414.
[24]Gu,M.(1992).Some moments concerning the Cox regression model.Report from the Dept.of Math.and Statist.McGill University 92-01.
[25]Gu,M.(1992).On the Edgworth expansion for U-statistics.Report from the Dept.of Math.and Statist.McGill University 92-03
[26]Gu,M.and Zheng,Z.(1993).On the Bartlett adjustment for the partial likelihood tatio test in the Cox regression model.Statistica Sinica 3 543,555.
[27]Hayakawa,T.(1991).The likelihood ratio criterion and the asymptotic expansion of its distribution.Annals of the Institute of Statistical Mathematics A 29 359-378.
[28]Jing,B.and Wood,T.A.(1996).Exponential empirical likelihood is not Bartlett correctable.The Annals of Statistics 24 365-369.
[29]Lai,T.L.and Wang,J.Q.(1993).Edgeworth expansion for symmetric statistics with appplicateons to bootstrap methods.Statistica Sinica 3 517-542.
[30]Lawley,D.N.(1956).A general method for approximating to the distribution of likelihood ratio criteria.Biometrika 71 233-244.
[31]Liu,Y.and Chen,J.(2009).Adjusted empirical likelihood with high-order precision.Manuscript.
[32]Owen,A.B.(1988).Empirical likelihood confidence interval for single functional.Biometrika 75 237-249.
[33]Owen,A.B.(1990).Empirical likelihood ratio confidence regions.Ann.Statist.18 90-120.
[34]Owen,A.B.(1991).Empirical likelihood for linear models.The Annals of Statistics 191725-1747.
[35]Owen,A.B.(2001).Empirical likelihood.Chapman and Hall/CRC,New York.
[36]Phillips,P.C.B.and Park,J.Y.(1988).On the formulation of Wald tests of nonlinear restrictions.Econometrica 56 1065-1083.
[37]Qin,G.and Jing,B.(2001).Empirical likelihood for cox regression model under random censorship.Communications in Statistics-Simulation and Computation 30 79-90.
[38]Qin,J.and Lawless,J.(1994).Empirical likelihood and general equations.Ann.Statist.22300-325.
[39]Schennach,S.M.(2007).Point estimation with exponentially tilted empirical likelihood.Ann.Statist.35 634-672.
[40]Sharp,S.A.(1975).Correction to likelihood ratio tests of hypotheses concerning the parameters of Markov chains.Biometrika 62 595-598.
[41]Strawderman,R.L.(1997).An asymptotic analysis of the logrank test.Lifetime Data Analysis 3 225-249.
[42]Strawderman,R.L.and Wells,M.T.(1997).Accurate bootstrap confidence limits for the cumulative hazard and survivor functions under random censoring.Journal of the American Statistical Association 92 1356-1374.
[43]Tableman,M.and Kim,J.S.(2002).Survival Analysis using S.Chapman and Hall/CRC,New York.
[44]Tsao,M.(2004).Bounds on coverage probabilities of the empirical likelihood ratio confidence regions.The Annals of Statistics 32 1215-1221.
[45]Tsiatis,A.A.(1981).A large sample study of the Cox's regression model.Ann.Statist.993-108.
[46]Tu,D.,Chen,J.,Shi,P.and Wu,Y.(2005).A Bartlett type correction for Rao's score test in cox regression model.Sankhy(?) 67 722-735.
[2]Andersen,P.K.,Borgan,O.,Gill,R.D.and Keiding,N.(1995).Statistical Models Based on Counting Processes.Springer,New York.
[3]Andersen,P.K.and Gill,D.R.(1982).Cox's regression model for counting processes.Ann.Statist.10 1100-1120.
[4]Baggerly,K.A.(1998).Empirical likelihood as a goodness-of-fit measure.Biometrika 85535-547.
[5]Barndorff-Nielsen,O.E.and Hall,P.(1988).On the level-error after Bartlett adjustment of the likelihood ratio statistic.Biometrika 75 374-378.
[6]Bartlett,M.S.(1937).Properties of sufficiency and statistical tests.Proceedings of the Royal Society A 160 268-282.
[7]Bhattacharya,R.N.and Ghosh,J.K.(1978).On the validity of the formal edgeworth expansion.Ann.Statist.6 434-451.
[8]Box,G.E.P.(1949).A general distribution theory for a class of likelihood criteria.Biometrika 36 317-346.
[9]Chandra,T.K.(1985).Asymptotic expansions of perturbed chi-square variables.Sankhy(?) A 47 100-110.
[10]Chandra,T.K.and Mukerjee,R.(1991).Bartlett-type modification of Rao's efficient score statistic.Journal Multivariate Analysis 36 103-112.
[11]Chen,J.,Variyath,A.M.and Abraham,B.(2008).Adjusted empirical likelihood and its properties.Journal of Computational and Graphical Statistics 17 1-18.
[12]Chesher,A.and Smith,R.(1994).Bartlett corrections to likelihood ratio tests.Unpublished paper,Department of Economics,University of Bristol.
[13]Corcoran,S.A.(1998).Bartlett adjustment of empirical discrepancy statistics.Biometrika 85 967-972.
[14]Cordeiro,G.M.(1987).On the corrections to the likelihood ratio statistics.Biometrika 74265-274.
[15]Cordeiro,G.M.and Ferrari,S.L.P.(1991).A modified score test statistic having chi-squared distribution to order n~(-1).Biometrika 78 573-582.
[16]Cox,D.R.(1972).Regression models and life tables(with discussion).J.Roy.Statist.Soc.Set.B.34 187-220.
[17]Cox,D.R.(1988).Some aspects of conditional and asymptotic inference:A review.Sankhy(?)A 50 314-337.
[18]Cribari-Neto,F.and Cordeiro,G.M.(1996).On Bartlett and Bartlett-type corrections.Econometric Reviews 15 339-367.
[19]DiCiccio,T.,Hall,P.and Romano,J.(1991).Empirical likelihood is Bartlett correctable.The Ann.Statist.19 1053-1061.
[20]Efron,B.(1981).Nonparametric standard errors and confidence intervals(with discussion).Canad.J.Statist.9 139-172.
[21]Ferrari,S.L.P.and Cribari-Neto,F.(1993).On the corrections to the Wald test of non-linear restrictions.Economics Letters 42 321-326.
[22]Fleming,T.R.and Harrington,D.P.(1991).Counting Processes and Survival Analysis.Wiley,New York.
[23]Gu,M.(1992).On the Edgeworth expansion and bootstrap approximation for the Cox regression model under random censorship.The Canadian Joural of Statistics 20 399-414.
[24]Gu,M.(1992).Some moments concerning the Cox regression model.Report from the Dept.of Math.and Statist.McGill University 92-01.
[25]Gu,M.(1992).On the Edgworth expansion for U-statistics.Report from the Dept.of Math.and Statist.McGill University 92-03
[26]Gu,M.and Zheng,Z.(1993).On the Bartlett adjustment for the partial likelihood tatio test in the Cox regression model.Statistica Sinica 3 543,555.
[27]Hayakawa,T.(1991).The likelihood ratio criterion and the asymptotic expansion of its distribution.Annals of the Institute of Statistical Mathematics A 29 359-378.
[28]Jing,B.and Wood,T.A.(1996).Exponential empirical likelihood is not Bartlett correctable.The Annals of Statistics 24 365-369.
[29]Lai,T.L.and Wang,J.Q.(1993).Edgeworth expansion for symmetric statistics with appplicateons to bootstrap methods.Statistica Sinica 3 517-542.
[30]Lawley,D.N.(1956).A general method for approximating to the distribution of likelihood ratio criteria.Biometrika 71 233-244.
[31]Liu,Y.and Chen,J.(2009).Adjusted empirical likelihood with high-order precision.Manuscript.
[32]Owen,A.B.(1988).Empirical likelihood confidence interval for single functional.Biometrika 75 237-249.
[33]Owen,A.B.(1990).Empirical likelihood ratio confidence regions.Ann.Statist.18 90-120.
[34]Owen,A.B.(1991).Empirical likelihood for linear models.The Annals of Statistics 191725-1747.
[35]Owen,A.B.(2001).Empirical likelihood.Chapman and Hall/CRC,New York.
[36]Phillips,P.C.B.and Park,J.Y.(1988).On the formulation of Wald tests of nonlinear restrictions.Econometrica 56 1065-1083.
[37]Qin,G.and Jing,B.(2001).Empirical likelihood for cox regression model under random censorship.Communications in Statistics-Simulation and Computation 30 79-90.
[38]Qin,J.and Lawless,J.(1994).Empirical likelihood and general equations.Ann.Statist.22300-325.
[39]Schennach,S.M.(2007).Point estimation with exponentially tilted empirical likelihood.Ann.Statist.35 634-672.
[40]Sharp,S.A.(1975).Correction to likelihood ratio tests of hypotheses concerning the parameters of Markov chains.Biometrika 62 595-598.
[41]Strawderman,R.L.(1997).An asymptotic analysis of the logrank test.Lifetime Data Analysis 3 225-249.
[42]Strawderman,R.L.and Wells,M.T.(1997).Accurate bootstrap confidence limits for the cumulative hazard and survivor functions under random censoring.Journal of the American Statistical Association 92 1356-1374.
[43]Tableman,M.and Kim,J.S.(2002).Survival Analysis using S.Chapman and Hall/CRC,New York.
[44]Tsao,M.(2004).Bounds on coverage probabilities of the empirical likelihood ratio confidence regions.The Annals of Statistics 32 1215-1221.
[45]Tsiatis,A.A.(1981).A large sample study of the Cox's regression model.Ann.Statist.993-108.
[46]Tu,D.,Chen,J.,Shi,P.and Wu,Y.(2005).A Bartlett type correction for Rao's score test in cox regression model.Sankhy(?) 67 722-735.
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