Generalized profiling estimation for global and adaptive penalized spline smoothing

We propose the generalized profiling method to estimate the multiple regression functions in the framework of penalized spline smoothing, where the regression functions and the smoothing parameter are estimated in two nested levels of optimization. The corresponding gradients and Hessian matrices ar...

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Veröffentlicht in:	Computational statistics & data analysis Jg. 53; H. 7; S. 2550 - 2562
Hauptverfasser:	Cao, Jiguo, Ramsay, James O.
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Amsterdam Elsevier B.V 15.05.2009 Elsevier
Schriftenreihe:	Computational Statistics & Data Analysis
Schlagworte:	Exact sciences and technology General topics Mathematics Multivariate analysis Numerical analysis Numerical analysis. Scientific computation Numerical methods in probability and statistics Parametric inference Probability and statistics Sciences and techniques of general use Statistics Variance estimation Optimization method Estimator robustness Adaptive estimation Penalty method Optimization Spline Variance Spline approximation Numerical approximation Smoothing parameter Regression function Regression spline Mathematical programming Data analysis Smoothing methods Covariance analysis Variance analysis Multiple regression Delta method Statistical regression Numerical analysis Statistical computation Simulation Smoothing Curvature
ISSN:	0167-9473, 1872-7352
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Abstract	We propose the generalized profiling method to estimate the multiple regression functions in the framework of penalized spline smoothing, where the regression functions and the smoothing parameter are estimated in two nested levels of optimization. The corresponding gradients and Hessian matrices are worked out analytically, using the Implicit Function Theorem if necessary, which leads to fast and stable computation. Our main contribution is developing the modified delta method to estimate the variances of the regression functions, which include the uncertainty of the smoothing parameter estimates. We further develop adaptive penalized spline smoothing to estimate spatially heterogeneous regression functions, where the smoothing parameter is a function that changes along with the curvature of regression functions. The simulations and application show that the generalized profiling method leads to good estimates for the regression functions and their variances.
AbstractList	We propose the generalized profiling method to estimate the multiple regression functions in the framework of penalized spline smoothing, where the regression functions and the smoothing parameter are estimated in two nested levels of optimization. The corresponding gradients and Hessian matrices are worked out analytically, using the Implicit Function Theorem if necessary, which leads to fast and stable computation. Our main contribution is developing the modified delta method to estimate the variances of the regression functions, which include the uncertainty of the smoothing parameter estimates. We further develop adaptive penalized spline smoothing to estimate spatially heterogeneous regression functions, where the smoothing parameter is a function that changes along with the curvature of regression functions. The simulations and application show that the generalized profiling method leads to good estimates for the regression functions and their variances.
Author	Ramsay, James O. Cao, Jiguo
Author_xml	– sequence: 1 givenname: Jiguo surname: Cao fullname: Cao, Jiguo email: jca76@sfu.ca organization: Department of Statistics and Actuarial Science, Simon Fraser University, Burnaby, BC, V5A1S6, Canada – sequence: 2 givenname: James O. surname: Ramsay fullname: Ramsay, James O. email: ramsay@psych.mcgill.ca organization: Department of Psychology, McGill University, 1205 Dr. Penfield Ave., Montreal, QC, H3A 1B1, Canada
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Cites_doi	10.1111/1467-842X.00119 10.2307/1270359 10.1111/j.2517-6161.1991.tb01837.x 10.1214/aos/1069362731 10.1111/j.1467-9868.2007.00610.x 10.1214/aos/1176347963 10.2307/2288926 10.1002/(SICI)1099-095X(199607)7:4<401::AID-ENV221>3.0.CO;2-D 10.2307/2291269 10.1111/j.2517-6161.1995.tb02034.x 10.1198/106186005X47345 10.2307/2291403 10.2307/1271131 10.2307/2291270 10.1214/aos/1176324704 10.1214/aos/1176349743 10.1007/s00180-007-0044-1 10.2307/2289875 10.2307/1390723 10.1111/j.2517-6161.1983.tb01239.x 10.1007/s001800000047 10.1007/BF02595870
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Keywords	Variance estimation Optimization method Estimator robustness Adaptive estimation Penalty method Optimization Spline Variance Spline approximation Numerical approximation Smoothing parameter Regression function Regression spline Mathematical programming Data analysis Smoothing methods Covariance analysis Variance analysis Multiple regression Delta method Statistical regression Numerical analysis Statistical computation Simulation Smoothing Curvature
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Title	Generalized profiling estimation for global and adaptive penalized spline smoothing
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