Uniform in bandwidth consistency of the kernel-type estimator of the Shannon's entropy

Salim Bouzebda; Issam Elhattab

doi:10.1016/j.crma.2009.12.007

Statistics

Uniform in bandwidth consistency of the kernel-type estimator of the Shannon's entropy
[Loi du logarithme uniforme pour un estimateur non paramétrique de l'entropie de Shannon]

Présenté par : Paul Deheuvels

Salim Bouzebda ¹ ; Issam Elhattab ¹

¹ L.S.T.A., Université Pierre et Marie Curie-Paris 6, 175, rue du Chevaleret, 8

Comptes Rendus. Mathématique, Volume 348 (2010) no. 5-6, pp. 317-321.

Résumé
Abstract

Dans cette Note, nous obtenons la consistance uniforme en terme de la fenêtre pour l'estimateur non paramétrique de l'entropie. Nos arguments de démonstration sont basés sur les résultats obtenus par Einmahl et Mason (2005) [10].

We establish uniform-in-bandwidth consistency for kernel-type estimators of the differential entropy. Our proofs rely on the methods of Einmahl and Mason (2005) [10].

Reçu le : 2008-07-22
Accepté le : 2009-12-02
Publié le : 2010-02-25

DOI : 10.1016/j.crma.2009.12.007

Affiliations des auteurs :

Salim Bouzebda ¹ ; Issam Elhattab ¹

¹ L.S.T.A., Université Pierre et Marie Curie-Paris 6, 175, rue du Chevaleret, 8

@article{CRMATH_2010__348_5-6_317_0,
     author = {Salim Bouzebda and Issam Elhattab},
     title = {Uniform in bandwidth consistency of the kernel-type estimator of the {Shannon's} entropy},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {317--321},
     publisher = {Elsevier},
     volume = {348},
     number = {5-6},
     year = {2010},
     doi = {10.1016/j.crma.2009.12.007},
     language = {en},
}

TY  - JOUR
AU  - Salim Bouzebda
AU  - Issam Elhattab
TI  - Uniform in bandwidth consistency of the kernel-type estimator of the Shannon's entropy
JO  - Comptes Rendus. Mathématique
PY  - 2010
SP  - 317
EP  - 321
VL  - 348
IS  - 5-6
PB  - Elsevier
DO  - 10.1016/j.crma.2009.12.007
LA  - en
ID  - CRMATH_2010__348_5-6_317_0
ER  -

%0 Journal Article
%A Salim Bouzebda
%A Issam Elhattab
%T Uniform in bandwidth consistency of the kernel-type estimator of the Shannon's entropy
%J Comptes Rendus. Mathématique
%D 2010
%P 317-321
%V 348
%N 5-6
%I Elsevier
%R 10.1016/j.crma.2009.12.007
%G en
%F CRMATH_2010__348_5-6_317_0

Salim Bouzebda; Issam Elhattab. Uniform in bandwidth consistency of the kernel-type estimator of the Shannon's entropy. Comptes Rendus. Mathématique, Volume 348 (2010) no. 5-6, pp. 317-321. doi : 10.1016/j.crma.2009.12.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.12.007/

Bibliographie
Cité par

[1] H. Akaike An approximation to the density function, Ann. Inst. Statist. Math., Tokyo, Volume 6 (1954), pp. 127-132

[2] J. Beirlant; E.J. Dudewicz; L. Györfi; E.C. van der Meulen Nonparametric entropy estimation: An overview, Int. J. Math. Stat. Sci., Volume 6 (1997) no. 1, pp. 17-39

[3] D. Bosq; J.-P. Lecoutre Théorie de l'estimation fonctionnelle. Économie et Statistiques Avancèes, Economica, Paris, 1987

[4] P. Deheuvels Uniform limit laws for kernel density estimators on possibly unbounded intervals, Bordeaux, 2000 (Stat. Ind. Technol.), Birkhäuser Boston, Boston, MA (2000), pp. 477-492

[5] P. Deheuvels; D.M. Mason General asymptotic confidence bands based on kernel-type function estimators, Stat. Inference Stoch. Process., Volume 7 (2004) no. 3, pp. 225-277

[6] L. Devroye; L. Györfi Nonparametric Density Estimation: The $L_{1}$ View, Wiley Series in Probability and Mathematical Statistics: Tracts on Probability and Statistics, John Wiley & Sons Inc., New York, 1985

[7] L. Devroye; G. Lugosi Combinatorial Methods in Density Estimation, Springer Series in Statistics, Springer-Verlag, New York, 2001

[8] L. Devroye; G.L. Wise Detection of abnormal behavior via nonparametric estimation of the support, SIAM J. Appl. Math., Volume 38 (1980) no. 3, pp. 480-488

[9] U. Einmahl; D.M. Mason An empirical process approach to the uniform consistency of kernel-type function estimators, J. Theoret. Probab., Volume 13 (2000) no. 1, pp. 1-37

[10] U. Einmahl; D.M. Mason Uniform in bandwidth consistency of kernel-type function estimators, Ann. Statist., Volume 33 (2005) no. 3, pp. 1380-1403

[11] E. Giné; D.M. Mason Uniform in bandwidth estimation of integral functionals of the density function, Scand. J. Statist., Volume 35 (2008) no. 4, pp. 739-761

[12] L. Györfi; E.C. van der Meulen On the nonparametric estimation of the entropy functional, Spetses, 1990 (NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.), Volume vol. 335, Kluwer Acad. Publ., Dordrecht (1991), pp. 81-95

[13] E. Parzen On estimation of a probability density function and mode, Ann. Math. Statist., Volume 33 (1962), pp. 1065-1076

[14] B.L.S. Prakasa Rao Nonparametric functional estimation, Probability and Mathematical Statistics, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1983

[15] M. Rosenblatt Remarks on some nonparametric estimates of a density function, Ann. Math. Statist., Volume 27 (1956), pp. 832-837

[16] C.E. Shannon A mathematical theory of communication, Bell System Tech. J., Volume 27 (1948), pp. 379-423 (623–656)

Cité par Sources :

Commentaires - Politique