Asymptotic Optimality of the Cₚ-Test for the Orthogonal Series Estimation of Regression

Previous article Next article Asymptotic Optimality of the Cₚ-Test for the Orthogonal Series Estimation of RegressionB. T. Polyak and A. B. TsybakovB. T. Polyak and A. B. Tsybakovhttps://doi.org/10.1137/1135037PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] N. N. Chentsov, A bound for an unknown distribution density in terms of the observations, Dokl. Akad. Nauk SSSR, 147 (1962), 45–48, (In Russian.) 26:837 Google Scholar[2] Yu. Sh. Verulava and , B. T. Polyak, Selection of the regression model order, Automat. Remote Control, 49 (1988), 1482–1494 0698.62065 Google Scholar[3] V. N. Vapnik, Recovering of Dependences from Empirical Data, Nauka, Moscow, 1979, (In Russian.) Google Scholar[4] I. I. Perel'man, Methodology of selecting the model structure for plant identification, Avtomat. i Telemekh., (1983), 5–29, (In Russian.) 85f:93072 Google Scholar[5] G. A. F. Seber, Linear regression analysis, John Wiley & Sons, New York-London-Sydney, 1977xvii+465 55:9428 0354.62055 Google Scholar[6] C. Mallows, Some comments on Cₚ, Technometrics, 15 (1973), 661–675 0269.62061 Google Scholar[7] Hirotugu Akaike, Statistical predictor identification, Ann. Inst. Statist. Math., 22 (1970), 203–217 44:3447 0259.62076 CrossrefGoogle Scholar[8] John Rice, Bandwidth choice for nonparametric regression, Ann. Statist., 12 (1984), 1215–1230 86c:62057 0554.62035 CrossrefGoogle Scholar[9] Wolfgang Härdle and , James Stephen Marron, Optimal bandwidth selection in nonparametric regression function estimation, Ann. Statist., 13 (1985), 1465–1481 87b:62047 0594.62043 CrossrefGoogle Scholar[10] Ker-Chau Li, Asymptotic optimality of C L and generalized cross-validation in ridge regression with application to spline smoothing, Ann. Statist., 14 (1986), 1101–1112 88c:62064 0629.62043 CrossrefGoogle Scholar[11] Peter Craven and , Grace Wahba, Smoothing noisy data with spline functions. Estimating the correct degree of smoothing by the method of generalized cross-validation, Numer. Math., 31 (1978/79), 377–403 81g:65018 0377.65007 CrossrefGoogle Scholar[12] I. A. Ibragimov and , R. Z. Khas'minskii, Nonparametric regression estimation, Dokl. Akad. Nauk SSSR, 252 (1980), 780–784, (In Russian.) 83b:62124 Google Scholar[13] M. S. Pinsker, Optimal filtration of square integrable signals in Gaussian noise, Problemy Peredachi Informatsii, 16 (1980), 52–68, (In Russian.) Google Scholar[14] S. Yu. Efroimovich, The adaptive algorithm of nonparametric regression, 1986, in Second IFAC Symp. Stock. Control., Vilnius, USSR. Preprints, Part II. Moscow, 112–114 Google Scholar[15] B. T. Polyak and , A. B. Tsybakov, Optimal projection estimates for a regression function of unknown smoothness, Soviet Math. Dokl., 39 (1989), 73–77 0704.62044 Google Scholar[16] Michel Loève, Probability theory, 2nd ed. The University Series in Higher Mathematics. D. Van Nostrand Co., Inc., Princeton, N. J.-Toronto-New York-London, 1960xvi+685 23:A670 0095.12201 Google Scholar[17] Ker-Chau Li, Asymptotic optimality for C p, C L, cross-validation and generalized cross-validation: discrete index set, Ann. Statist., 15 (1987), 958–975 89c:62112 0653.62037 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails On improvability of model selection by model averagingJournal of Econometrics, Vol. 229, No. 2 | 1 Aug 2022 Cross Ref Weighted validation of heteroscedastic regression models for better selectionStatistical Analysis and Data Mining: The ASA Data Science Journal, Vol. 15, No. 1 | 17 August 2021 Cross Ref High-Dimensional Adaptive Minimax Sparse Estimation With InteractionsIEEE Transactions on Information Theory, Vol. 65, No. 9 | 1 Sep 2019 Cross Ref Trigonometric regression estimation for observations with additive and multiplicative errorsCommunications in Statistics - Theory and Methods, Vol. 45, No. 3 | 1 April 2015 Cross Ref Cross-validation for selecting a model selection procedureJournal of Econometrics, Vol. 187, No. 1 | 1 Jul 2015 Cross Ref Statistical inference for the optimal approximating modelProbability Theory and Related Fields, Vol. 155, No. 3-4 | 22 February 2012 Cross Ref Inverse Problems in StatisticsInverse Problems and High-Dimensional Estimation | 5 May 2011 Cross Ref Model SelectionHandbook of Financial Time Series | 10 February 2009 Cross Ref Minimal Penalties for Gaussian Model SelectionProbability Theory and Related Fields, Vol. 138, No. 1-2 | 4 July 2006 Cross Ref PREDICTION/ESTIMATION WITH SIMPLE LINEAR MODELS: IS IT REALLY THAT SIMPLE?Econometric Theory, Vol. 23, No. 01 | 6 December 2006 Cross Ref A new algorithm for fixed design regression and denoisingAnnals of the Institute of Statistical Mathematics, Vol. 56, No. 3 | 1 Sep 2004 Cross Ref Oracle inequalities for inverse problemsThe Annals of Statistics, Vol. 30, No. 3 | 1 Jun 2002 Cross Ref Random rates in anisotropic regression (with a discussion and a rejoinder by the authors)The Annals of Statistics, Vol. 30, No. 2 | 1 Apr 2002 Cross Ref Empirical Processes Techniques for the Spectral Estimation of Fractional ProcessesEmpirical Process Techniques for Dependent Data | 1 Jan 2002 Cross Ref Nonparametric estimation of transfer functions: rates of convergence and adaptationIEEE Transactions on Information Theory, Vol. 44, No. 2 | 1 Mar 1998 Cross Ref An asymptotic property of model selection criteriaIEEE Transactions on Information Theory, Vol. 44, No. 1 | 1 Jan 1998 Cross Ref From Model Selection to Adaptive EstimationFestschrift for Lucien Le Cam | 1 Jan 1997 Cross Ref Adaptive hypothesis testing using waveletsThe Annals of Statistics, Vol. 24, No. 6 | 1 Dec 1996 Cross Ref A Family of Asymptotically Optimal Methods for Choosing the Order of a Projective Regression EstimateB. T. Polyak and A. B. TsybakovTheory of Probability & Its Applications, Vol. 37, No. 3 | 17 July 2006AbstractPDF (912 KB) Volume 35, Issue 2| 1991Theory of Probability & Its Applications205-409 History Submitted:01 December 1987Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1135037Article page range:pp. 293-306ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics