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## Séminaire de Statistique

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Organisateurs : Mélisande Albert, Dominique Bontemps, Pierre Neuvial

Jour et lieu habituels : le mardi à 11h00 en salle 106 (bâtiment 1R1).

• ### Mardi 18 décembre 11:00-12:00 - Vincent Feuillard - Airbus

A multivariate extreme value theory approach to anomaly clustering and visualization

Lieu : Salle 106, Bat 1R1 - Motivated by a wide variety of applications from fraud detection to aviation safety management, unsupervised anomaly detection is the subject of much attention in the machine-learning literature. We developed novel statistical techniques for tackling anomaly detection borrowing concepts and tools from machine-learning and multivariate extreme value analysis both at the same time. Usually, anomaly detection algorithms declared extremes as anomaly, whereas all extremes values are not anomalies. We study the dependance structure of rare events in the context of high dimensional and propose an algorithm to detect this structure under a sparse assumption. This approach can reduce drastically the false alarm rate : anomalies then correspond to the observation of simultaneous very large/extreme values for groups of variables that have not been identified yet. A data-driven methodology for learning the sparse representation of extreme behaviours has been developed in Goix (2016). An advantage of this method lies in its straightforward interpretability. In addition, the representation of the dependance structure in the extremes thus designed induces a specific notion of (dis-)similarity among anomalies, that paves the way for elaborating visualization tools for operators in the spirit of those proposed for large graphs. We also describe how this approach applies to functional data collected for aircraft safety purposes after an appropriate preliminary filtering stage.

• ### Mardi 8 janvier 2019 11:15-12:15 - Gérard Letac - IMT

Les quantiles d’une famille exponentielle

Résumé : Soit $P$ une probabilité sur $R$ et $P_t (dx)=e^{xt} P(dx)/L(t)$. Il est facile de voir que $t$ est la moyenne de $P_t$ pour tout $t$ si et seulement si $P$ est gaussienne. C'est beaucoup moins aisé si on remplace le mot moyenne par le mot médiane, voire le mot quantile. Nous traitons aussi le cas analogue des lois gamma (voir ArXiv 1810-11917). Ceci utilise le résultat de Choquet Deny de 1960 qui dit que si $H$ est une densité de probabilité et si $f$ est positive alors $f=f*H$ si et seulement si $f$ est barycentre des $x\mapsto e^{xt}$ tels que $\int e^{xt}H(x)dx$. En collaboration avec Mauro Piccioni et Bartosz Kolodziejek.

Lieu : Salle 106, Bat 1R1

• ### Mardi 15 janvier 2019 11:15-12:15 - Nicolas Bousquet - Sorbonne Université

Séminaire de Statistique

Lieu : Salle 106, Bat 1R1

• ### Mardi 22 janvier 2019 09:15-10:45 - Marc Hallin (Séminaire commun proba-stats) - ECARES et Département de Mathématique Université libre de Bruxelles

Center-Outward Distribution Functions, Quantiles, Ranks, and Signs in R^d : A Measure Transportation Approach

Résumé : Unlike the real line, the $d$-dimensional space $R^d$, for $d \geq 2$, is not canonically ordered. As a consequence, such fundamental and strongly order-related univariate concepts as quantile and distribution functions, and their empirical counterparts, involving ranks and signs, do not canonically extend to the multivariate context. Palliating that lack of a canonical ordering has remained an open problem for more than half a century, and has generated an abundant literature, motivating, among others, the development of statistical depth and copula-based methods. We show that, unlike the many definitions that have been proposed in the literature, the measure transportation-based ones introduced in Chernozhukov, Galichon, Hallin and Henry (2017) enjoy all the properties (distribution-freeness and the maximal invariance property that entails preservation of semiparametric efficiency) that make univariate quantiles and ranks successful tools for semiparametric statistical inference. We therefore propose a new \it center-outward definition of multivariate distribution and quantile functions, along with their empirical counterparts, for which we establish a Glivenko-Cantelli result---the quintessential property of all distribution functions. Our approach, based on results by McCann (1995), is geometric rather than analytical and, contrary to the Monge-Kantorovich one in Chernozhukov et al. (2017) (which assumes compact supports, hence finite moments of all orders), does not require any moment assumptions. The resulting ranks and signs are shown to be strictly distribution-free, and maximal invariant under the action of a data-driven class of (order-preserving) transformations generating the family of absolutely continuous distributions ; that maximal invariance, in view of a general result by Hallin and Werker (2003), is the theoretical foundation of the semiparametric efficiency preservation property of ranks. The corresponding quantiles are equivariant under the same transformations.

Lieu : Amphi Schwartz

• ### Mardi 29 janvier 2019 11:15-12:15 - Stéphane Chrétien - National Physical Laboratory

Séminaire de Statistique

Lieu : Salle 106, Bat 1R1

• ### Mardi 5 février 2019 11:15-12:15 - Emilie Lebarbier - AgroParisTech/INRA

Séminaire de Statistique

Lieu : Salle 106, Bat 1R1

• ### Mardi 12 février 2019 11:15-12:15 - Tristan Mary-Huard - AgroParisTech/INRA

Séminaire de Statistique

Lieu : Salle 106, Bat 1R1

• ### Mardi 19 février 2019 11:15-12:15 - Bruno Pelletier - Université Rennes II

Séminaire de Statistique

Lieu : Salle 106, Bat 1R1

• ### Mardi 2 avril 2019 09:15-10:45 - Patricia Reynaud-Bouret - Université de Nice Sophia-Antipolis

Séminaire commun Proba-Stat

Lieu : amphi Schwartz

iCal