Why minimax is not that pessimistic

Aurelia Fraysse

ESAIM: Probability and Statistics (2013)

  • Volume: 17, page 472-484
  • ISSN: 1292-8100

Abstract

top
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.

How to cite

top

Fraysse, Aurelia. "Why minimax is not that pessimistic." ESAIM: Probability and Statistics 17 (2013): 472-484. <http://eudml.org/doc/273625>.

@article{Fraysse2013,
abstract = {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.},
author = {Fraysse, Aurelia},
journal = {ESAIM: Probability and Statistics},
keywords = {minimax theory; maxiset theory; Besov spaces; prevalence; wavelet bases},
language = {eng},
pages = {472-484},
publisher = {EDP-Sciences},
title = {Why minimax is not that pessimistic},
url = {http://eudml.org/doc/273625},
volume = {17},
year = {2013},
}

TY - JOUR
AU - Fraysse, Aurelia
TI - Why minimax is not that pessimistic
JO - ESAIM: Probability and Statistics
PY - 2013
PB - EDP-Sciences
VL - 17
SP - 472
EP - 484
AB - 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.
LA - eng
KW - minimax theory; maxiset theory; Besov spaces; prevalence; wavelet bases
UR - http://eudml.org/doc/273625
ER -

References

top
  1. [1] F. Autin, Point de vue maxiset en estimation non paramétrique. Ph.D. thesis, Université Paris 7 (2004). 
  2. [2] Y. Benyamini and J. Lindenstrauss, Geometric nonlinear functional analysis, Colloquium Publications, vol. 1. American Mathematical Society (AMS) (2000). Zbl0946.46002MR1727673
  3. [3] L. Birgé, Approximation dans les espaces métriques et théorie de l’estimation. Z. Wahrscheinlichkeitstheorie Verw. Gebiete65 (1983) 181–237. Zbl0506.62026MR722129
  4. [4] J.P.R. Christensen, On sets of Haar measure zero in Abelian Polish groups. Isr. J. Math.13 (1972) 255–260. Zbl0249.43002MR326293
  5. [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. Zbl0997.62025MR1848302
  6. [6] I. Daubechies, Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math.41 (1988) 909–996. Zbl0644.42026MR951745
  7. [7] P. Dodos, Dichotomies of the set of test measures of a Haar-null set. Isr. J. Math.144 (2004) 15–28. Zbl1062.43002MR2121532
  8. [8] D. Donoho and I. Johnstone, Minimax risk over lp-balls for lq-error. Probab. Theory Relat. Fields99 (1994) 277–303. Zbl0802.62006MR1278886
  9. [9] D. Donoho and I. Johnstone, Minimax estimation via wavelet shrinkage. Ann. Stat.26 (1998) 879–921. Zbl0935.62041MR1635414
  10. [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. Zbl0891.62025MR1462946
  11. [11] A. Fraysse, Generic validity of the multifractal formalism. SIAM J. Math. Anal.37 (2007) 593–607. Zbl1132.28320MR2338422
  12. [12] B. Hunt, The prevalence of continuous nowhere differentiable function. Proc. Am. Math. Soc.122 (1994) 711–717. Zbl0861.26003MR1260170
  13. [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. Zbl0763.28009MR1161274
  14. [14] I.A. Ibragimov and R.Z. Hasminski, Statistical estimation, Applications of Mathematics, vol. 16. Springer-Verlag (1981). Zbl0467.62026MR620321
  15. [15] S. Jaffard, Old friends revisited: The multifractal nature of some classical functions. J. Fourier Anal. Appl.3 (1997) 1–22. Zbl0880.28007MR1428813
  16. [16] S. Jaffard, On the Frisch-Parisi conjecture. J. Math. Pures Appl.79 (2000) 525–552. Zbl0963.28009MR1770660
  17. [17] G. Kerkyacharian and D. Picard, Density estimation by kernel and wavelets methods: optimality of Besov spaces. Stat. Probab. Lett.18 (1993) 327–336. Zbl0793.62019MR1245704
  18. [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. Zbl1107.62323MR1821645
  19. [19] G. Kerkyacharian and D. Picard, Minimax or maxisets? Bernoulli8 (2002) 219–253. Zbl1006.62005MR1895892
  20. [20] S. Mallat, A wavelet tour of signal processing. Academic Press, San Diego, CA (1998) xxiv. Zbl0998.94510MR1614527
  21. [21] Y. Meyer, Ondelettes et opérateurs. Hermann (1990). Zbl0694.41037MR1085487
  22. [22] A.S. Nemirovskiĭ, B.T. Polyak and A.B. Tsybakov, The rate of convergence of nonparametric estimates of maximum likelihood type. Problemy Peredachi Informatsii21 (1985) 17–33. Zbl0616.62048MR820705
  23. [23] M.S. Pinsker, Optimal filtration of square-integrable signals in Gaussian noise. Probl. Infor. Transm.16 (1980) 52–68. Zbl0452.94003MR624591
  24. [24] V. Rivoirard, Maxisets for linear procedures, Stat. Probab. Lett.67 (2004) 267–275. Zbl1125.60307MR2053529
  25. [25] V. Rivoirard, Nonlinear estimation over weak Besov spaces and minimax Bayes method, Bernoulli12 (2006) 609–632. Zbl1125.62001MR2248230
  26. [26] E. Stein, Singular integrals and differentiability properties of functions. Princeton University Press (1970). Zbl0207.13501MR290095
  27. [27] A. Tsybakov, Introduction to nonparametric estimation. Springer Series in Statistics, Springer, New York (2009). Zbl1029.62034MR2724359
  28. [28] A. Van der Vaart, Asymptotic statistics, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 3. Cambridge University Press (1998). Zbl0910.62001MR1652247

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.