指数族非线性模型的统计诊断及其试验设计
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摘要
本文首先对均值漂移模型建立几何结构，给出了该模型的两类曲率立体阵，由此推出了参数的最大似然估计量的二阶展开式，进而得到了基于曲率的诊断统计量，然后推得带线性约束指数族非线性模型的基于统计曲率的二阶诊断统计量。再计算模型的回归参数β在1-α置信水平下的置信域的体积，为了便于计算，需把置信域投影到切空间中去，投影后的置信域是一个椭球体，用Taylor展开方法对此椭球体的体积进行二阶近似，推出原参数置信域的体积的近似表达式，然后对回归模型进行试验设计，根据使置信域体积最小准则，在模型函数的设计变量的定义域上确定点列，选取使置信域体积达到最小的设计点。
This paper first builds the geometric framework for the mean shift model, then gives two kinds of the curvature arrays and gets the second order expression of the MLE of the parameter, and then carries out the expression of the diagnostic statistic with the curvature array, and also carries out the second-order diagnostic statistic for the models with linear restriction. Second we calculate the volume of the confidence region of the parameters β
with a confidence level 1 - α , in order to make the calculating easier, it is necessary to map the confidence region into tangent space, and the confidence region, which, after having been mapped, will present itself as an ellipsoid. We use the Taylor series expansion method to get the second-order approximate expression of the confidence region of the parameters, and then according to the principle of minimizing the volume of the confidence region, this paper carries out experimental design for the regression model, and then fixes the point range in the domain of definition of design variable for the model function so as to select the designing point, which can minimize the volume of the confidence region.
This paper first builds the geometric framework for the mean shift model, then gives two kinds of the curvature arrays and gets the second order expression of the MLE of the parameter, and then carries out the expression of the diagnostic statistic with the curvature array, and also carries out the second-order diagnostic statistic for the models with linear restriction. Second we calculate the volume of the confidence region of the parameters β
with a confidence level 1 - α , in order to make the calculating easier, it is necessary to map the confidence region into tangent space, and the confidence region, which, after having been mapped, will present itself as an ellipsoid. We use the Taylor series expansion method to get the second-order approximate expression of the confidence region of the parameters, and then according to the principle of minimizing the volume of the confidence region, this paper carries out experimental design for the regression model, and then fixes the point range in the domain of definition of design variable for the model function so as to select the designing point, which can minimize the volume of the confidence region.
引文
1. A.I.Khuri, A Note on D-Optimal Designs for Partially Nonlinear Regression Models. Technometrics February 1984
2. Anthony C.Atkinson and William G..Hunter, The Design of Experiments for Parameter Estimation. Technometrics, May 1968
3. Bates, Watts. Relative Curvature Measure of Nonlinearlity. J.R..Statist.Soc.B, 1980(42), 1-25
4. Cook, R.D. and Tsai, C.L. Diagnostics for Assessing the Accuracy of Normal Approximations in Exponential Family Nonlinear Models. JASA, 1990(85):770-777
5. Cook, Weisberg. Residual and Influence in Regression. New York, Chapman and Hall 1982
6. Cordeiro and Paula. Improved Likelihood Ratio Statistics for Exponential Family Nonlinear Models. Biometrika, 1989(76): 93-100
7. David C.Hamilton and Donald GWatts, A Quadratic Design Criterion for Precise Estimation in Nonlinear Regression Models. Technometrics, August 1985
8. McCullagh, P. and Nelder, J.A. Generalized Linear Models. London: Chapman & Hall, 1989
9. Peter D.H.Hill, D-Optimal Designs for Partially Nonlinear Regression Models. Technometrics May 1980
10. R.C.St.John and N.R.Draper, D-Optimal for Regression Designs: A Review Technometrics February 1975
11. V.V.Fedorov, Theory of Optimal Experiments. Academic Press New York and London 1972
12. Wei Bocheng Exponential Family Nonlinear Models Springer 1998
13. Wei Bocheng Geometry of Exponential Family Nonlinear Models and Asymptotic Inference. Science in China Ser . 1995, 298-308
14. Wei Bocheng Some Second Order Asymptotics in Nonlinear Regression. Aust. J. Statist. 1991, 75-84
15. Wei Bocheng, Ma Yangming. Parametric Estimators and Influence Analysis for Nonlinear Exponential Family Regression Model. Journal of Southeast University. 1992, 22(6)
16. Wei Bocheng, Shi, J. Q. Statistical Models in Regression Diagnostics. Ann. Inst. Statist. Math. 1994, 267-278
17. Zhu Zhongyi. Asymptotics of MLE in Exponential Family Nonlinear Models Journal of Southeast University. 1995, 25(6)
18．陈希孺．数理统计引论．北京：科学出版社 1981
19．陈希孺，王松桂．近代回归分析．合肥：安徽教育出版社 1987
20．程云鹏，张凯院，徐仲．矩阵论．西安：西北工业大学出版社 1998
21．方开泰，马长兴．正交与均匀试验设计．北京：科学出版社2001
22．梁之舜．概率论及数理统计．北京：高等教育出版社 1988
23．茆诗松等．回归分析及其试验设计．上海：华东师范大学出版社1981，255—302
24．茆诗松，王静龙，濮晓龙．高等数理统计．北京：高等教育出版社 1998
25．茆诗松，周纪芗，陈颖．试验设计．北京：中国统计出版社2004
26．牛长山，徐通模．试验设计与数据处理．西安：西安交通大学出版社1988
27．韦博成．近代非线性回归分析．南京：东南大学出版社1989
28．韦博成．统计诊断引论．南京：东南大学出版社 1990
29．韦博成．统计推断与微分几何．北京：清华大学出版社 1988
30．韦博成，马阳明．指数型回归模型参数置信域的曲率表示．工程数学学报1995，12(1)
31．韦博成，万方焕．指数族非线性模型置信域的曲率表示．东南大学学报1996，26(3)
32．韦博成 万方焕等译．非线性回归分析及其应用．北京：中国统计出版社1997
33．王沫然，肖劲松．MATLAB 5.X与科学计算．北京：清华大学出版社2000
34．王松桂，陈敏，陈立萍．线性统计模型．北京：高等教育出版社 1999
35．王万中，茆诗松，曾林蕊．试验的设计与分析．北京：高等教育出版社2004
36．韦博成 马阳明．带约束非线性回归模型的影响分析．南京：东南大学学报1993，23(3)
37．朱玲妹．广义非线性模型的诊断与影响分析．南京经济学院学报．1995，46-55。
38．赵慧秀．带有线性约束的指数族非线性模型的基于曲率的统计分析．南京理工大学学报．2003
39．张世英，李铁山．非线性回归的诊断问题．天津大学学报．1998，257—264
40．朱仲义，韦博成．半参数非线性模型的统计诊断与影响分析．应用数学学报．2001，24(4)，568—581。
2. Anthony C.Atkinson and William G..Hunter, The Design of Experiments for Parameter Estimation. Technometrics, May 1968
3. Bates, Watts. Relative Curvature Measure of Nonlinearlity. J.R..Statist.Soc.B, 1980(42), 1-25
4. Cook, R.D. and Tsai, C.L. Diagnostics for Assessing the Accuracy of Normal Approximations in Exponential Family Nonlinear Models. JASA, 1990(85):770-777
5. Cook, Weisberg. Residual and Influence in Regression. New York, Chapman and Hall 1982
6. Cordeiro and Paula. Improved Likelihood Ratio Statistics for Exponential Family Nonlinear Models. Biometrika, 1989(76): 93-100
7. David C.Hamilton and Donald GWatts, A Quadratic Design Criterion for Precise Estimation in Nonlinear Regression Models. Technometrics, August 1985
8. McCullagh, P. and Nelder, J.A. Generalized Linear Models. London: Chapman & Hall, 1989
9. Peter D.H.Hill, D-Optimal Designs for Partially Nonlinear Regression Models. Technometrics May 1980
10. R.C.St.John and N.R.Draper, D-Optimal for Regression Designs: A Review Technometrics February 1975
11. V.V.Fedorov, Theory of Optimal Experiments. Academic Press New York and London 1972
12. Wei Bocheng Exponential Family Nonlinear Models Springer 1998
13. Wei Bocheng Geometry of Exponential Family Nonlinear Models and Asymptotic Inference. Science in China Ser . 1995, 298-308
14. Wei Bocheng Some Second Order Asymptotics in Nonlinear Regression. Aust. J. Statist. 1991, 75-84
15. Wei Bocheng, Ma Yangming. Parametric Estimators and Influence Analysis for Nonlinear Exponential Family Regression Model. Journal of Southeast University. 1992, 22(6)
16. Wei Bocheng, Shi, J. Q. Statistical Models in Regression Diagnostics. Ann. Inst. Statist. Math. 1994, 267-278
17. Zhu Zhongyi. Asymptotics of MLE in Exponential Family Nonlinear Models Journal of Southeast University. 1995, 25(6)
18．陈希孺．数理统计引论．北京：科学出版社 1981
19．陈希孺，王松桂．近代回归分析．合肥：安徽教育出版社 1987
20．程云鹏，张凯院，徐仲．矩阵论．西安：西北工业大学出版社 1998
21．方开泰，马长兴．正交与均匀试验设计．北京：科学出版社2001
22．梁之舜．概率论及数理统计．北京：高等教育出版社 1988
23．茆诗松等．回归分析及其试验设计．上海：华东师范大学出版社1981，255—302
24．茆诗松，王静龙，濮晓龙．高等数理统计．北京：高等教育出版社 1998
25．茆诗松，周纪芗，陈颖．试验设计．北京：中国统计出版社2004
26．牛长山，徐通模．试验设计与数据处理．西安：西安交通大学出版社1988
27．韦博成．近代非线性回归分析．南京：东南大学出版社1989
28．韦博成．统计诊断引论．南京：东南大学出版社 1990
29．韦博成．统计推断与微分几何．北京：清华大学出版社 1988
30．韦博成，马阳明．指数型回归模型参数置信域的曲率表示．工程数学学报1995，12(1)
31．韦博成，万方焕．指数族非线性模型置信域的曲率表示．东南大学学报1996，26(3)
32．韦博成 万方焕等译．非线性回归分析及其应用．北京：中国统计出版社1997
33．王沫然，肖劲松．MATLAB 5.X与科学计算．北京：清华大学出版社2000
34．王松桂，陈敏，陈立萍．线性统计模型．北京：高等教育出版社 1999
35．王万中，茆诗松，曾林蕊．试验的设计与分析．北京：高等教育出版社2004
36．韦博成 马阳明．带约束非线性回归模型的影响分析．南京：东南大学学报1993，23(3)
37．朱玲妹．广义非线性模型的诊断与影响分析．南京经济学院学报．1995，46-55。
38．赵慧秀．带有线性约束的指数族非线性模型的基于曲率的统计分析．南京理工大学学报．2003
39．张世英，李铁山．非线性回归的诊断问题．天津大学学报．1998，257—264
40．朱仲义，韦博成．半参数非线性模型的统计诊断与影响分析．应用数学学报．2001，24(4)，568—581。