. Démonstration, La preuve est empruntéè a van der Vaart et Wellner, 1996.

. Démonstration, On se place, sans perte de généralité, dans le cas o` u C est constituée de parties C ? {x 1

C. Si and . E. Est-héréditaire-(-i, alors le résultat est immédiat. Le but est maintenant de montrer que l'on peut toujours se rameneràramenerà une classe héréditaire

. Si-x-i-/-?-c, C, et onôteonôte x i ` a C seulement si cela crée une nouvelle partie dans C. Nous allons montrer que si T i (C) vérifie le résultat du lemme, alors C aussi

?. Si-x-i and ?. A. , (car il existe b ? A tel que a ? {x i } = b) On en déduit que x i ? T i (C), et, par suite, que T i (C) = C. Ainsi, a ? {x i } est picked out par C

|. De, En combinant cettedernì eréeré egalité et (A.11), il vient ? C ({x 1 Ainsi, le lemme est vérifié par C s'il est vérifié par T i (C)

C. Comme and . |c|, C, il existe un entier ? tel que T n,?+1 (C) = T n,? (C) De plus, ? ? C?C |C|. Soit alors D = T n,? (C) Pour tout 1 ? i ? n

. Finalement, le résultatrésultatétant vérifié par T n,? (C), il l'est aussi par C

A. Le-résultat-de-la-proposition, 2.2 est maintenant immédiat En effet, une classe C d'index V (C) ne pulvérisant aucun ensemble de V (C) points, tous les ensembles de points pulvérisés par C font partie, des ensembles de cardinal au plus V (C)?1. Or il y a exactement n C ?V (C), p.1

. Dans-ce-paragrapheles-construire, Le premier principe sera, dans les deux cas, de se rameneràramenerà une classe « simple » et en obtenir la propriété V C. Ensuite, la stabilité par union, intersection et produit cartésien (cf. les propositions présentées ci-après) des classes V C permet de construire des classes plus « complexes ». Notons cependant que ce premier principe n'estévidemmentestévidemment pas suffisant pour générer l'ensemble des classes V C, ou obtenir la propriété V C de l'ensemble des classes V C, 1984.

V. Classes, Une classe C de sous-ensembles d'un ensemble X présente un index V (C) = 0 (ou demanì eré equivalente S(C) = ?1) si et seulement si C est vide, De plus, V (C) = 1 (ou demanì eré equivalente S(C) = 0) si et seulement si C ne contient qu'un ensemble. Ainsi, S(C) ? 1 si et seulement si C contient au moins deux ensembles

. Démonstration and . Au-moins-un-ensemble, Par ailleurs, si C contient au moins deux ensembles, alors il existe A, B dans C et x ? X tels que x ? A \ B. Donc C pulvérise {x}, et S(C) ? 1. Réciproquement, si S(C) ? 1, alors C contient au moins deux ensembles

. Démonstration, Compte tenu de la proposition précédente, on a dans les deux cas S(C) ? 1

. Si, C est linéairement ordonné pour l'inclusion, soit {x, y} un ensemble pulvérisé par C. On sélectionne A, B ? C tels que A ? {x, y} = {x} et B ? {x, y} = {y}

S. L. De-même, ensembles de C sont disjoints, en raisonnant comme dans le cas précédent, et en prenant C ? C tel que {x, y} = C ? {x, y}, l'hypothèse C et A disjoints (ou C et B disjoints) conduitàconduità une contradiction

. La-théorie-développée and . Vapnik, Il existe plusieurs définitions, nonéquivalentesnonéquivalentes, des classes V C de fonctions. Nous n'´ evoquerons ici que les classes V C dites d'hypographes (pour les définitions des classes de fonctions V C dites major et hull, nous renvoyons notammentàtammentà van der Vaart et Wellner La définition la plus intéressante, dans notre cadre de travail, est celle des classes V C d'hypographes, puisque cesdernì eres présentent la propriété de nombre de recouvrement polynomial, 1996.

. Enfin, hypographes sont incluses dans l'ensemble des classes présentant un nombre de recouvrement polynomial. Perspectives de recherche directes de ce type d'estimateurs est la construction de scores de risque de pathologie, autrement dit l'estimation de la probabilité de développer une maladié etant donné un ensemble de

. Viallon, les médecins, mais aussi le grand public sont de plus en plus demandeurs d'outils statistiques permettant de prédire des risque absolus individuels de maladie . Dans cette optique, et en collaboration nous avons entrepris l'´ elaboration d'un score de risque de cancer du sein pour les femmes françaises, A.7 Constructions de scores de risques de pathologies Les chercheurs Françoise Clavel-Chepelon ( ´ Equipe ERI20-INSERM) et Jacques Bénichou (CHU Rouen) Pour ce faire, nous avons appliqué un modèle de Cox, avecâgeavecâge enéchelleenéchelle de temps, p.3, 2006.

. Viallon, Différentes méthodes sont communément utilisées dans lesétudeslesétudes de validation de tels modèles. Cependant, elles présentent généralement un biais (dû principalementàprincipalementà la présence de censures) Dans ce travail, nous listons les méthodes existantes, soulignons leurs défauts respectifs de façon théorique, puis proposons notre méthode. Des simulations ainsi qu'un exemple de validation, des scores de risques, 2006.

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P. Deheuvels and D. Et-mason, A tail empirical process approach to some nonstandard laws of the iterated logarithm, Journal of Theoretical Probability, vol.2, issue.1, pp.53-85, 1991.
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P. Deheuvels and D. Et-mason, Functional Laws of the Iterated Logarithm for the Increments of Empirical and Quantile Processes, The Annals of Probability, vol.20, issue.3, pp.1248-1287, 1992.
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P. Deheuvels and D. Et-mason, Functional Laws of the Iterated Logarithm for Local Empirical Processes Indexed by Sets, The Annals of Probability, vol.22, issue.3, pp.1619-1661, 1994.
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P. Deheuvels and D. Et-mason, Nonstandard local empirical processes indexed by sets, Journal of Statistical Planning and Inference, vol.45, issue.1-2, pp.91-112, 1995.
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P. Deheuvels and D. Et-mason, General Asymptotic Confidence Bands Based on Kernel-type Function Estimators, Statistical Inference for Stochastic Processes, vol.7, issue.3, pp.225-277, 2004.
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P. Deheuvels and D. Et-mason, Estimating conditional distributions under random censorship, 2006.

M. Delecroix, O. Lopez, and V. Et-patilea, Nonlinear censored regression using synthetic data. Preprint, 2006.
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S. Derbort, H. Dette, and A. Et-munk, A test for additivity in nonparametric regression, Annals of the Institute of Statistical Mathematics, vol.54, issue.1, pp.60-82, 2002.
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R. Dudley, Uniform central limit theorems, 1999.

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P. Eggermont and V. Et-la-riccia, Maximum penalized likelihood estimation : Density estimation, 2001.

U. Einmahl and D. Et-mason, An empirical process approach to the uniform consistency of kernel type estimators, Journal of Theoretical Probability, vol.13, issue.1, pp.1-13, 2000.
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U. Einmahl and D. M. Et-mason, Uniform in bandwidth consistency of kernel-type function estimators, The Annals of Statistics, vol.33, issue.3, pp.1380-1403, 2005.
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J. Fan and I. Et-gijbels, Censored Regression: Local Linear Approximations and their Applications, Journal of the American Statistical Association, vol.8, issue.426, pp.560-570, 1994.
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J. Fan and I. Et-gijbels, Local polynomial modelling and its application, 1996.
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A. Földes and L. Et-rejt?-o, A LIL type result for the product limit estimator, Zeitschrift f??r Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol.32, issue.1, pp.75-86, 1981.
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J. H. Friedman and W. Et-stuetzle, Projection Pursuit Regression, Journal of the American Statistical Association, vol.4, issue.376, pp.817-823, 1981.
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E. Giné and A. Et-guillou, Laws of the iterated logarithm for censored data, pp.2042-2067, 1999.

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L. Györfi, M. Kohler, A. Krzyzak, and H. Et-walk, A distribution-free theory of nonparametric regression, 2002.
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J. Hajek, Z. Sidak, and P. Et-sen, Theory of Rank Tests, 1999.

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W. Härdle, Applied nonparametric regression, 1990.

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A. Thiebaut and J. Et-bénichou, Choice of time-scale in Cox's model analysis of epidemiologic cohort data: a simulation study, Statistics in Medicine, vol.6, issue.2, pp.3803-3820, 2004.
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