Why minimax is not that pessimistic
ESAIM: Probability and Statistics, Tome 17 (2013), pp. 472-484.

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

DOI : 10.1051/ps/2012002
Classification : 62C20, 28C20, 46E35
Mots clés : minimax theory, maxiset theory, Besov spaces, prevalence, wavelet bases
@article{PS_2013__17__472_0,
     author = {Fraysse, Aurelia},
     title = {Why minimax is not that pessimistic},
     journal = {ESAIM: Probability and Statistics},
     pages = {472--484},
     publisher = {EDP-Sciences},
     volume = {17},
     year = {2013},
     doi = {10.1051/ps/2012002},
     mrnumber = {3070887},
     zbl = {1284.62091},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps/2012002/}
}
TY  - JOUR
AU  - Fraysse, Aurelia
TI  - Why minimax is not that pessimistic
JO  - ESAIM: Probability and Statistics
PY  - 2013
SP  - 472
EP  - 484
VL  - 17
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ps/2012002/
DO  - 10.1051/ps/2012002
LA  - en
ID  - PS_2013__17__472_0
ER  - 
%0 Journal Article
%A Fraysse, Aurelia
%T Why minimax is not that pessimistic
%J ESAIM: Probability and Statistics
%D 2013
%P 472-484
%V 17
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ps/2012002/
%R 10.1051/ps/2012002
%G en
%F PS_2013__17__472_0
Fraysse, Aurelia. Why minimax is not that pessimistic. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 472-484. doi : 10.1051/ps/2012002. http://www.numdam.org/articles/10.1051/ps/2012002/

[1] F. Autin, Point de vue maxiset en estimation non paramétrique. Ph.D. thesis, Université Paris 7 (2004).

[2] Y. Benyamini and J. Lindenstrauss, Geometric nonlinear functional analysis, Colloquium Publications, vol. 1. American Mathematical Society (AMS) (2000). | MR | Zbl

[3] L. Birgé, Approximation dans les espaces métriques et théorie de l'estimation. Z. Wahrscheinlichkeitstheorie Verw. Gebiete 65 (1983) 181-237. | MR | Zbl

[4] J.P.R. Christensen, On sets of Haar measure zero in Abelian Polish groups. Isr. J. Math. 13 (1972) 255-260. | MR | Zbl

[5] A. Cohen, R. Devore, G. Kerkyacharian and D. Picard, Maximal spaces with given rate of convergence for thresholding algorithms. Appl. Comput. Harmon. Anal. 11 (2001) 167-191. | MR | Zbl

[6] I. Daubechies, Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. 41 (1988) 909-996. | MR | Zbl

[7] P. Dodos, Dichotomies of the set of test measures of a Haar-null set. Isr. J. Math. 144 (2004) 15-28. | MR | Zbl

[8] D. Donoho and I. Johnstone, Minimax risk over lp-balls for lq-error. Probab. Theory Relat. Fields 99 (1994) 277-303. | MR | Zbl

[9] D. Donoho and I. Johnstone, Minimax estimation via wavelet shrinkage. Ann. Stat. 26 (1998) 879-921. | MR | Zbl

[10] 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. | MR | Zbl

[11] A. Fraysse, Generic validity of the multifractal formalism. SIAM J. Math. Anal. 37 (2007) 593-607. | MR | Zbl

[12] B. Hunt, The prevalence of continuous nowhere differentiable function. Proc. Am. Math. Soc. 122 (1994) 711-717. | MR | Zbl

[13] B. Hunt, T. Sauer and J. Yorke, Prevalence: a translation invariant “almost every” on infinite dimensional spaces. Bull. Am. Math. Soc. 27 (1992) 217-238. | MR | Zbl

[14] I.A. Ibragimov and R.Z. Hasminski, Statistical estimation, Applications of Mathematics, vol. 16. Springer-Verlag (1981). | MR | Zbl

[15] S. Jaffard, Old friends revisited: The multifractal nature of some classical functions. J. Fourier Anal. Appl. 3 (1997) 1-22. | MR | Zbl

[16] S. Jaffard, On the Frisch-Parisi conjecture. J. Math. Pures Appl. 79 (2000) 525-552. | MR | Zbl

[17] G. Kerkyacharian and D. Picard, Density estimation by kernel and wavelets methods: optimality of Besov spaces. Stat. Probab. Lett. 18 (1993) 327-336. | MR | Zbl

[18] G. Kerkyacharian and D. Picard, Thresholding algorithms, maxisets and well-concentrated bases. Test 9 (2000) 283-344, With comments, and a rejoinder by the authors. | MR | Zbl

[19] G. Kerkyacharian and D. Picard, Minimax or maxisets? Bernoulli 8 (2002) 219-253. | MR | Zbl

[20] S. Mallat, A wavelet tour of signal processing. Academic Press, San Diego, CA (1998) xxiv. | MR | Zbl

[21] Y. Meyer, Ondelettes et opérateurs. Hermann (1990). | MR | Zbl

[22] A.S. Nemirovskiĭ, B.T. Polyak and A.B. Tsybakov, The rate of convergence of nonparametric estimates of maximum likelihood type. Problemy Peredachi Informatsii 21 (1985) 17-33. | MR | Zbl

[23] M.S. Pinsker, Optimal filtration of square-integrable signals in Gaussian noise. Probl. Infor. Transm. 16 (1980) 52-68. | MR | Zbl

[24] V. Rivoirard, Maxisets for linear procedures, Stat. Probab. Lett. 67 (2004) 267-275. | MR | Zbl

[25] V. Rivoirard, Nonlinear estimation over weak Besov spaces and minimax Bayes method, Bernoulli 12 (2006) 609-632. | MR | Zbl

[26] E. Stein, Singular integrals and differentiability properties of functions. Princeton University Press (1970). | MR | Zbl

[27] A. Tsybakov, Introduction to nonparametric estimation. Springer Series in Statistics, Springer, New York (2009). | MR | Zbl

[28] A. Van Der Vaart, Asymptotic statistics, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 3. Cambridge University Press (1998). | MR | Zbl

Cité par Sources :