Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-28T05:41:04.475Z Has data issue: false hasContentIssue false
  • English
  • Français

Why minimax is not that pessimistic∗

Published online by Cambridge University Press:  03 June 2013

Aurelia Fraysse
Aurelia Fraysse*
Affiliation:
L2S, SUPELEC, CNRS, University Paris-Sud, 3 rue Joliot-Curie, 91190 Gif-Sur-Yvette, France. [email protected]
Get access

Abstract

In nonparametric statistics a classical optimality criterion for estimation procedures isprovided by the minimax rate of convergence. However this point of view can be subject tocontroversy as it requires to look for the worst behavior of an estimation procedure in agiven space. The purpose of this paper is to introduce a new criterion based on genericbehavior of estimators. We are here interested in the rate of convergence obtained withsome classical estimators on almost every, in the sense of prevalence, function in a Besovspace. We also show that generic results coincide with minimax ones in these cases.

Keywords

Minimax theorymaxiset theoryBesov spacesprevalencewavelet bases
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 472 - 484
Copyright
© EDP Sciences, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

F. Autin, Point de vue maxiset en estimation non paramétrique. Ph.D. thesis, Université Paris 7 (2004).
Y. Benyamini and J. Lindenstrauss, Geometric nonlinear functional analysis, Colloquium Publications, vol. 1. American Mathematical Society (AMS) (2000).
Birgé, L., Approximation dans les espaces métriques et théorie de l’estimation. Z. Wahrscheinlichkeitstheorie Verw. Gebiete 65 (1983) 181–237. Google Scholar
Christensen, J.P.R., On sets of Haar measure zero in Abelian Polish groups. Isr. J. Math. 13 (1972) 255–260. Google Scholar
Cohen, A., DeVore, R., Kerkyacharian, G. and Picard, D., Maximal spaces with given rate of convergence for thresholding algorithms. Appl. Comput. Harmon. Anal. 11 (2001) 167–191. Google Scholar
Daubechies, I., Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. 41 (1988) 909–996. Google Scholar
Dodos, P., Dichotomies of the set of test measures of a Haar-null set. Isr. J. Math. 144 (2004) 15–28. Google Scholar
Donoho, D. and Johnstone, I., Minimax risk over l p-balls for l q-error. Probab. Theory Relat. Fields 99 (1994) 277–303. Google Scholar
Donoho, D. and Johnstone, I., Minimax estimation via wavelet shrinkage. Ann. Stat. 26 (1998) 879–921. Google Scholar
D.L. Donoho, I.M. Johnstone, G. Kerkyacharian and D. Picard, Universal near minimaxity of wavelet shrinkage. Festschrift for Lucien Le Cam, Springer, New York (1997) 183–218.
Fraysse, A., Generic validity of the multifractal formalism. SIAM J. Math. Anal. 37 (2007) 593–607. Google Scholar
Hunt, B., The prevalence of continuous nowhere differentiable function. Proc. Am. Math. Soc. 122 (1994) 711–717. Google Scholar
Hunt, B., Sauer, T. and Yorke, J., Prevalence: a translation invariant “almost every” on infinite dimensional spaces. Bull. Am. Math. Soc. 27 (1992) 217–238. Google Scholar
I.A. Ibragimov and R.Z. Hasminski, Statistical estimation, Applications of Mathematics, vol. 16. Springer-Verlag (1981).
Jaffard, S., Old friends revisited: The multifractal nature of some classical functions. J. Fourier Anal. Appl. 3 (1997) 1–22. Google Scholar
Jaffard, S., On the Frisch-Parisi conjecture. J. Math. Pures Appl. 79 (2000) 525–552. Google Scholar
Kerkyacharian, G. and Picard, D., Density estimation by kernel and wavelets methods: optimality of Besov spaces. Stat. Probab. Lett. 18 (1993) 327–336. Google Scholar
Kerkyacharian, G. and Picard, D., Thresholding algorithms, maxisets and well-concentrated bases. Test 9 (2000) 283–344, With comments, and a rejoinder by the authors. Google Scholar
Kerkyacharian, G. and Picard, D., Minimax or maxisets? Bernoulli 8 (2002) 219–253. Google Scholar
S. Mallat, A wavelet tour of signal processing. Academic Press, San Diego, CA (1998) xxiv.
Y. Meyer, Ondelettes et opérateurs. Hermann (1990).
Nemirovskiĭ, A.S., Polyak, B.T. and Tsybakov, A.B., The rate of convergence of nonparametric estimates of maximum likelihood type. Problemy Peredachi Informatsii 21 (1985) 17–33. Google Scholar
Pinsker, M.S., Optimal filtration of square-integrable signals in Gaussian noise. Probl. Infor. Transm. 16 (1980) 52–68. Google Scholar
Rivoirard, V., Maxisets for linear procedures, Stat. Probab. Lett. 67 (2004) 267–275. Google Scholar
Rivoirard, V., Nonlinear estimation over weak Besov spaces and minimax Bayes method, Bernoulli 12 (2006) 609–632. Google Scholar
E. Stein, Singular integrals and differentiability properties of functions. Princeton University Press (1970).
A. Tsybakov, Introduction to nonparametric estimation. Springer Series in Statistics, Springer, New York (2009).
A. Van der Vaart, Asymptotic statistics, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 3. Cambridge University Press (1998).