广义非线性模型的Bayes估计
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摘要
针对广义非线性模型的参数估计问题,提出了从参数的条件后验分布中抽取观测值来估计参数值的Bayes估计法.利用贝叶斯统计分析中蒙特卡洛抽样方法中的M-H算法和Gibbs抽样算法相结合的混合算法进行分析,通过参数的条件后验分布抽取出每次迭代时的参数值,并利用参数的样本路径图和均值遍历图验证迭代时马尔科夫链的收敛性;计算马尔科夫链达到收敛后参数的后验均值得到参数的Bayes估计;通过对产品销售数据的实证分析,比较Bayes估计和极大似然估计的偏差,验证M-H算法和Gibbs抽样算法在对广义非线性模型的参数进行Bayes估计时的简洁性、有效性以及可行性.
For the problem of parameter estimation in generalized nonlinear models,a method of Bayesian estimation is proposed to extract the observations from the conditional posterior distribution of the parameters to estimate the parameters. In the Bayesian statistical analysis,the hybrid algorithm of the M-H algorithm and the Gibbs sampling algorithm in the Monte Carlo sampling method is used to analyze the model. The parameters values are extracted through the conditional posterior distribution of the parameters at each iteration,and the convergence of the Markov chain at iteration is verified by using the sample path figure and the mean traverse figure of the parameters. The Bayes estimation of parameter is obtained by calculating the posterior mean value of the Markov chain after the chain achieves convergence. Through the empirical analysis of product sales data,the biases of Bayesian estimation and Maximum Likelihood estimation are compared to verify the simplicity,validity,and feasibility of the M-H algorithm and the Gibbs sampling algorithm for Bayesian estimation of the parameters of the generalized nonlinear model.
For the problem of parameter estimation in generalized nonlinear models,a method of Bayesian estimation is proposed to extract the observations from the conditional posterior distribution of the parameters to estimate the parameters. In the Bayesian statistical analysis,the hybrid algorithm of the M-H algorithm and the Gibbs sampling algorithm in the Monte Carlo sampling method is used to analyze the model. The parameters values are extracted through the conditional posterior distribution of the parameters at each iteration,and the convergence of the Markov chain at iteration is verified by using the sample path figure and the mean traverse figure of the parameters. The Bayes estimation of parameter is obtained by calculating the posterior mean value of the Markov chain after the chain achieves convergence. Through the empirical analysis of product sales data,the biases of Bayesian estimation and Maximum Likelihood estimation are compared to verify the simplicity,validity,and feasibility of the M-H algorithm and the Gibbs sampling algorithm for Bayesian estimation of the parameters of the generalized nonlinear model.
引文
[1]韦博成,林金官,解锋昌.统计诊断[M].北京:高等教育出版社,2009WEI B C,LIN J G,XIE F C. Statistical Diagnosis[M].Beijing:Higher Education Press,2009
[2]牛翔宇,冯予.缺失数据下广义非线性回归的经验似然及诊断[J].重庆工商大学学报(自然科学版),2016,33(6):15—21NIU X Y,FENG Y. Empirical Likelihood and Diagnosis of Generalized Nonlinear Regression in Missing Data[J].Journal of Chongqing Technology and Business University(Natural Science Edition),2016,33(6):15—21
[3] XU W J,LIN J G. Diagnostics in Generalized Nonlinear Models Based on Maximum L Q-Likelihood Estimation[J].Journal of Southeast University,2013,29(1):106—110
[4]韦来生.贝叶斯统计[M].北京:高等教育出版社,2016WEI L S. Bayesian Statistics[M]. Beijing:Higher Education Press,2016
[5]王明高.贝叶斯统计模型及其应用研究[M].北京:经济科学出版社,2017WANG M G. Bayesian Statistical Model and Its Application Research[M]. Beijing:Economic Science Press,2017
[6]和燕,彭燕梅,唐年胜.带不可忽略缺失数据的再生散度随机效应的Bayes估计[J].宁夏大学学报(自然科学版),2011,32(3):193—197HE Y,PENG Y M,TANG N S. Bayes Estimation of Regenerated Divergence Random Effect Model with Nonnegligible Data[J]. Journal of Ningxia University(Natural Science Edition),2011,32(3):193—197
[7]唐年胜,邱世芳.非线性再生散度模型的Bayes估计[J].数理统计与管理,2007,26(6):999—1006TANG N S,QIU S F. Bayes Estimation of Nonlinear Regenerative Divergence model[J]. Application of Statistics and Management,2007,26(6):999—1006
[8]左艳芳,刘伟.非线性再生散度模型两种估计方法的比较[J].云南民族大学学报(自然科学版),2008,17(4):320—324ZUO Y F,LIU W. Comparison of Two Estimation Methods for Nonlinear Regenerative Divergence Model[J]. Journal of Yunnan Nationalities University(Natural Sciences Edition),2008,17(4):320—324
[9] LEE S Y,TANG N S. Bayesian Analysis of Structural Equation Models with Mixed Exponential Family and Ordered Categorical Data[J]. British Journal of Mathematical and Statistical,2006,59:151—172
[10] THANOON T Y,ADNAN R. Analysis of Generalized Nonlinear Structural Equation Models by Using Bayesian Approach with Application[J]. Pakistan Journal of Statistics and Operation Research,2017,13(1):17—45
[2]牛翔宇,冯予.缺失数据下广义非线性回归的经验似然及诊断[J].重庆工商大学学报(自然科学版),2016,33(6):15—21NIU X Y,FENG Y. Empirical Likelihood and Diagnosis of Generalized Nonlinear Regression in Missing Data[J].Journal of Chongqing Technology and Business University(Natural Science Edition),2016,33(6):15—21
[3] XU W J,LIN J G. Diagnostics in Generalized Nonlinear Models Based on Maximum L Q-Likelihood Estimation[J].Journal of Southeast University,2013,29(1):106—110
[4]韦来生.贝叶斯统计[M].北京:高等教育出版社,2016WEI L S. Bayesian Statistics[M]. Beijing:Higher Education Press,2016
[5]王明高.贝叶斯统计模型及其应用研究[M].北京:经济科学出版社,2017WANG M G. Bayesian Statistical Model and Its Application Research[M]. Beijing:Economic Science Press,2017
[6]和燕,彭燕梅,唐年胜.带不可忽略缺失数据的再生散度随机效应的Bayes估计[J].宁夏大学学报(自然科学版),2011,32(3):193—197HE Y,PENG Y M,TANG N S. Bayes Estimation of Regenerated Divergence Random Effect Model with Nonnegligible Data[J]. Journal of Ningxia University(Natural Science Edition),2011,32(3):193—197
[7]唐年胜,邱世芳.非线性再生散度模型的Bayes估计[J].数理统计与管理,2007,26(6):999—1006TANG N S,QIU S F. Bayes Estimation of Nonlinear Regenerative Divergence model[J]. Application of Statistics and Management,2007,26(6):999—1006
[8]左艳芳,刘伟.非线性再生散度模型两种估计方法的比较[J].云南民族大学学报(自然科学版),2008,17(4):320—324ZUO Y F,LIU W. Comparison of Two Estimation Methods for Nonlinear Regenerative Divergence Model[J]. Journal of Yunnan Nationalities University(Natural Sciences Edition),2008,17(4):320—324
[9] LEE S Y,TANG N S. Bayesian Analysis of Structural Equation Models with Mixed Exponential Family and Ordered Categorical Data[J]. British Journal of Mathematical and Statistical,2006,59:151—172
[10] THANOON T Y,ADNAN R. Analysis of Generalized Nonlinear Structural Equation Models by Using Bayesian Approach with Application[J]. Pakistan Journal of Statistics and Operation Research,2017,13(1):17—45