Comptes Rendus
no. 12
Probability Theory
A functional Hungarian construction for the sequential empirical process
[Une construction hongroise fonctionnelle pour le processus empirique séquentiel]

Présenté par : Michel Talagrand

Michael Jähnisch1 ; Michael Nussbaum2
1 SAP AG, Neurottstrasse 16, 69190 Walldorf, Germany
2 Department of Mathematics, Cornell University, Ithaca, NY 14853, USA
Comptes Rendus. Mathématique, Volume 341 (2005) no. 12, pp. 761-763.

We establish a KMT coupling for the sequential empirical process and the Kiefer–Müller process. The processes are indexed by functions f from a Hölder class H, but the supremum over f∈H is taken outside the probability. Compared to the coupling in sup-norm, this allows for larger functional classes H. The result is useful for proving asymptotic equivalence of certain nonparametric statistical experiments.

Nous établissons un couplage KMT pour le processus empirique séquentiel et le processus de Kiefer–Müller. Les processus sont indexés par des fonctions f appartenant à une classe de Hölder H, mais le supremum est pris en dehors de la probabilité. Comparé au couplage en norme sup, ceci permet des classes de fonctions H plus larges. Le résultat peut être utilisé pour démontrer l'équivalence asymptotique de certaines expériences statistiques non-paramétriques.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2005.10.022

Michael Jähnisch 1 ; Michael Nussbaum 2

1 SAP AG, Neurottstrasse 16, 69190 Walldorf, Germany
2 Department of Mathematics, Cornell University, Ithaca, NY 14853, USA 
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Michael Jähnisch; Michael Nussbaum. A functional Hungarian construction for the sequential empirical process. Comptes Rendus. Mathématique, Volume 341 (2005) no. 12, pp. 761-763. doi : 10.1016/j.crma.2005.10.022. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.10.022/

[1] J. Bretagnolle; P. Massart Hungarian constructions from the non-asymptotic viewpoint, Ann. Probab., Volume 17 (1989), pp. 239-256

[2] N. Castelle; F. Laurent-Bonvalot Strong approximation of bivariate uniform empirical processes, Ann. Inst. H. Poincaré Probab. Statist., Volume 34 (1998), pp. 425-480

[3] R. Dudley Real Analysis and Probability, Wadsworth Brooks/Cole, Pacific Grove, CA, 1989

[4] I. Grama; M. Nussbaum A functional Hungarian construction for sums of independent random variables, Ann. Inst. H. Poincaré Probab. Statist., Volume 38 (2002), pp. 923-957

[5] I. Grama; M. Nussbaum Asymptotic equivalence for nonparametric generalized linear models, Probab. Theory Related Fields, Volume 111 (1998), pp. 167-214

[6] I. Grama; M. Nussbaum Asymptotic equivalence for nonparametric regression, Math. Methods Statist., Volume 11 (2002) no. 1, pp. 1-36

[7] M. Jähnisch, Asymptotische Äquivalenz für ein Modell unabhängiger nicht identisch verteilter Daten, Doctoral Thesis, Humboldt Universität zu Berlin, 1999

[8] M. Jähnisch; M. Nussbaum Asymptotic equivalence for a model of independent non identically distributed observations, Statistics and Decisions, Volume 21 (2003) no. 3, pp. 197-218

[9] V. Koltchinskii Komlós–Major–Tusnády approximation for the general empirical process and Haar expansions of classes of functions, J. Theoret. Probab., Volume 7 (1994) no. 1, pp. 73-118

[10] J. Komlós; P. Major; G. Tusnády An approximation of partial sums of independent r.v.'s and the sample df. I, Z. Wahr. Verw. Gebiete, Volume 32 (1975), pp. 111-131

[11] M. Nussbaum Asymptotic equivalence of density estimation and Gaussian white noise, Ann. Statist., Volume 24 (1996), pp. 2399-2430

[12] E. Rio Local invariance principles and their application to density estimation, Probab. Theory Related Fields, Volume 98 (1994), pp. 21-45

[13] G. Shorack; J. Wellner Empirical Processes with Applications to Statistics, Wiley, New York, 1986

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