Advertisement

Minimal Penalties for Gaussian Model Selection

  • Lucien Birgé
  • Pascal Massart
Article

Abstract

This paper is mainly devoted to a precise analysis of what kind of penalties should be used in order to perform model selection via the minimization of a penalized least-squares type criterion within some general Gaussian framework including the classical ones. As compared to our previous paper on this topic (Birgé and Massart in J. Eur. Math. Soc. 3, 203–268 (2001)), more elaborate forms of the penalties are given which are shown to be, in some sense, optimal. We indeed provide more precise upper bounds for the risk of the penalized estimators and lower bounds for the penalty terms, showing that the use of smaller penalties may lead to disastrous results. These lower bounds may also be used to design a practical strategy that allows to estimate the penalty from the data when the amount of noise is unknown. We provide an illustration of the method for the problem of estimating a piecewise constant signal in Gaussian noise when neither the number, nor the location of the change points are known.

Keywords

Gaussian linear regression Variable selection Model selection Mallows’ Cp Penalized least-squares 

Mathematics Subject Classification (2000)

Primary 62G05 Secondary 62G07 62J05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abramovich, F., Benjamini, Y., Donoho, D.L., Johnstone, I.M.: Adapting to unknown sparsity by controlling the false discovery rate. Ann. Statist. 34, (2006)Google Scholar
  2. 2.
    Akaike H. (1969). Statistical predictor identification. Ann. Inst. Statist. Math. 22:203–217CrossRefMathSciNetGoogle Scholar
  3. 3.
    Akaike H. (1973). Information theory and an extension of the maximum likelihood principle. In: Petrov P.N., Csaki F. (eds) Proceedings 2nd International Symposium on Information Theory. Akademia Kiado, Budapest, pp. 267–281Google Scholar
  4. 4.
    Akaike H. (1974). A new look at the statistical model identification. IEEE Trans. Autom. Control 19:716–723zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Akaike H. A Bayesian analysis of the minimum AIC procedure. Ann. Inst. Statist. Math. 30, Part A, 9–14 (1978)Google Scholar
  6. 6.
    Amemiya T. (1985). Advanced Econometrics. Basil Blackwell, OxfordGoogle Scholar
  7. 7.
    Barron A.R., Birgé L., Massart P. (1999). Risk bounds for model selection via penalization. Probab. Theory Relat. Fields 113:301–415zbMATHCrossRefGoogle Scholar
  8. 8.
    Barron A.R., Cover T.M. (1991). Minimum complexity density estimation. IEEE Trans. Inf. Theory 37:1034–1054CrossRefMathSciNetGoogle Scholar
  9. 9.
    Birgé, L.: An alternative point of view on Lepski’s method. In: de Gunst, M.C.M., Klaassen, C.A.J., van der Vaart, A.W. (eds.) State of the Art in Probability and Statistics, Festschrift for Willem R. van Zwet, Institute of Mathematical Statistics, Lecture Notes–Monograph Series, Vol. 36. 113–133 (2001)Google Scholar
  10. 10.
    Birgé L., Massart P. (1998). Minimum contrast estimators on sieves: exponential bounds and rates of convergence. Bernoulli 4:329–375zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Birgé L., Massart P. (2001). Gaussian model selection. J. Eur. Math. Soc. 3:203–268zbMATHCrossRefGoogle Scholar
  12. 12.
    Birgé, L., Massart, P.: A generalized C p criterion for Gaussian model selection. Technical Report No 647. Laboratoire de Probabilités, Université Paris VI (2001) http://www.proba. jussieu.fr/mathdoc/preprints/index.html#2001Google Scholar
  13. 13.
    Daniel C., Wood F.S. (1971). Fitting Equations to Data. Wiley, New YorkzbMATHGoogle Scholar
  14. 14.
    Draper N.R., Smith H. (1981). Applied Regression Analysis, 2nd edn. Wiley, New YorkzbMATHGoogle Scholar
  15. 15.
    Efron B., Hastie R., Johnstone I.M., Tibshirani R. (2004). Least angle regression. Ann. Statist. 32:407–499zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Feller W. (1968). An Introduction to Probability Theory and its Applications, Vol I (3rd edn). Wiley, New YorkGoogle Scholar
  17. 17.
    George E.I., Foster D.P. (2000). Calibration and empirical Bayes variable selection. Biometrika 87:731–747zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Gey S., Nédélec E. (2005). Model selection for CART regression trees. IEEE Trans. Inf. Theory 51:658–670CrossRefGoogle Scholar
  19. 19.
    Guyon X., Yao J.F. (1999). On the underfitting and overfitting sets of models chosen by order selection criteria. Jour. Multivar. Anal. 70:221–249zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Hannan E.J., Quinn B.G. (1979). The determination of the order of an autoregression. J.R.S.S., B 41:190–195zbMATHMathSciNetGoogle Scholar
  21. 21.
    Hoeffding W. (1963). Probability inequalities for sums of bounded random variables. J.A.S.A. 58:13–30zbMATHMathSciNetGoogle Scholar
  22. 22.
    Hurvich K.L., Tsai C.-L. (1989). Regression and time series model selection in small samples. Biometrika 76:297–307zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Johnstone, I.: Chi-square oracle inequalities. In: de Gunst, M.C.M., Klaassen, C.A.J. van der Vaart, A.W. (eds.) State of the Art in Probability and Statistics, Festschrift for Willem R. van Zwet, Institute of Mathematical Statistics, Lecture Notes–Monograph Series, Vol. 36. pp. 399–418 (2001)Google Scholar
  24. 24.
    Kneip A. (1994). Ordered linear smoothers. Ann. Statist. 22:835–866zbMATHMathSciNetGoogle Scholar
  25. 25.
    Lavielle M., Moulines E. (2000). Least Squares estimation of an unknown number of shifts in a time series. J. Time Series Anal. 21:33–59zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Lebarbier E. (2005). Detecting multiple change-points in the mean of a Gaussian process by model selection. Signal Proces. 85:717–736CrossRefGoogle Scholar
  27. 27.
    Li K.C. (1987). Asymptotic optimality for C p, C L, cross-validation, and generalized cross-validation: Discrete index set. Ann. Statist. 15:958–975zbMATHMathSciNetGoogle Scholar
  28. 28.
    Loubes, J.-M., Massart, P.: Discussion of “Least angle regression” by Efron, B., Hastie, R., Johnstone, I., Tibshirani, R. Ann. Statist. 32, 460–465 (2004).Google Scholar
  29. 29.
    Mallows C.L. (1973). Some comments on C p. Technometrics 15:661–675zbMATHCrossRefGoogle Scholar
  30. 30.
    Massart P. (1990). The tight constant in the D.K.W. inequality. Ann. Probab. 18:1269–1283zbMATHMathSciNetGoogle Scholar
  31. 31.
    McQuarrie A.D.R., Tsai C.-L. (1998). Regression and Time Series Model Selection. World Scientific, SingaporezbMATHGoogle Scholar
  32. 32.
    Mitchell T.J., Beauchamp J.J. (1988). Bayesian variable selection in linear regression. J.A.S.A. 83:1023–1032zbMATHMathSciNetGoogle Scholar
  33. 33.
    Polyak B.T., Tsybakov A.B. (1990). Asymptotic optimality of the C p-test for the orthogonal series estimation of regression. Theory Probab. Appl. 35:293–306zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Rissanen J. (1978). Modeling by shortest data description. Automatica 14:465–471zbMATHCrossRefGoogle Scholar
  35. 35.
    Schwarz G. (1978). Estimating the dimension of a model. Ann. Statist. 6:461–464zbMATHMathSciNetGoogle Scholar
  36. 36.
    Shen X., Ye J. (2002). Adaptive model selection. J.A.S.A. 97:210–221zbMATHMathSciNetGoogle Scholar
  37. 37.
    Shibata R. (1981). An optimal selection of regression variables. Biometrika 68:45–54zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Wallace D.L. (1959). Bounds on normal approximations to Student’s and the chi-square distributions. Ann. Math. Stat. 30:1121–1130MathSciNetGoogle Scholar
  39. 39.
    Whittaker E.T., Watson G.N. (1927). A Course of Modern Analysis. Cambridge University Press, LondonzbMATHGoogle Scholar
  40. 40.
    Yang Y. (2005). Can the strenghths of AIC and BIC be shared? A conflict between model identification and regression estimation. Biometrika 92:937–950CrossRefMathSciNetGoogle Scholar
  41. 41.
    Yao Y.C. (1988). Estimating the number of change points via Schwarz criterion. Stat. Probab. Lett. 6:181–189zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.UMR 7599 “Probabilités et modèles aléatoires”, Laboratoire de Probabilités, boîte 188Université Paris VIParis Cedex 05France
  2. 2.UMR 8628 “Laboratoire de Mathématiques”, Bât. 425Université Paris SudOrsay CedexFrance

Personalised recommendations