## Section: New Results

### Semi and non parametric methods

#### Modelling extremal events

Participants : Stéphane Girard, Laurent Gardes.

This is joint work with Cécile Amblard (TimB in TIMC laboratory, Univ. Grenoble 1), Myriam Garrido (INRA Clermont-Ferrand), Armelle Guillou (Univ. Strasbourg), and Jean Diebolt (CNRS, Univ. Marne-la-vallée).

Our first achievement is the development of new estimators: kernel estimators and bias correction through exponential regression [16] . Our second achievement is the construction of a goodness-of-fit test for the distribution tail. Usual tests are not adapted to this problem since they essentially check the adequation to the central part of the distribution. Next, we aim at adapting extreme-value estimators to take into account covariate information. Such estimators would include extreme conditional quantiles estimators, which are closely linked to the frontier estimators presented in Section 6.3.2 . Finally, more future work will include the study of multivariate extreme values. To this aim, a research on some particular copulas [1] , [38] has been initiated with Cécile Amblard, since they are the key tool for building multivariate distributions [62] .

#### Boundary estimation

Participants : Stéphane Girard, Laurent Gardes.

This is joint work with Anatoli Iouditski (Univ. Joseph Fourier, Grenoble), Guillaume Bouchard (Xerox, Meylan), Pierre Jacob and Ludovic Menneteau (Univ. Montpellier 2) and Alexandre Nazin (IPU, Moscow, Russia).

Two different and complementary approaches are developped.

##### Extreme quantiles approach.

Here, the boundary bounding the set of points is viewed as the larger level set of the points distribution. This is then an extreme quantile curve estimation problem. We propose estimators based on projection as well as on kernel regression methods applied on the extreme values set [57] , [56] , [55] , [58] , for particular set of points. In this framework, we can obtain the asymptotic distribution of the error between estimators and the true frontier [53] ,[22] . Our future work will be to define similar methods based on wavelets expansions in order to estimate non-smooth boundaries, and on local polynomials estimators to get rid of boundary effects. Besides, we are also working on the extension of our results to more general sets of points. This work has been initiated in the PhD work of Laurent Gardes [48] , co-directed by Pierre Jacob and Stéphane Girard and in [31] with the consideration of star-shaped supports.

##### Linear programming approach.

Here, the boundary of a set of points is defined has a
closed curve bounding all the points and with smallest associate
surface. It is thus natural to reformulate the boundary estimation
method as a linear programming problem [41] , [40] , [54] .
The resulting estimate is parsimonious, it only relies on a small number of
points. This method belongs to the Support Vector Machines (SVM)
techniques. Their finite sample performances are very impressive
but their asymptotic properties are not very well known,
the difficulty being that there is no explicit formula of the estimator.
However, such properties are of great interest, in particular to reduce the
estimator bias. Two directions of research will be investigated. The first one
consists in modifying the optimization problem itself. The second one is to
use *Jacknife* like methods, combining two biased estimators so that the
two bias cancel out.
One of the goals of our work is also to establish
the speed of convergence of such methods
in order to try to improve them.

#### Modelling nuclear plants

Participants : Laurent Gardes, Stéphane Girard.

This is joint work with Nadia Perot, Nicolas Devictor and Michel Marquès (CEA).

One of the main activities of the Laboratoire de Conduite et Fiabilité des Réacteurs (CEA Cadarache) concerns the probabilistic analysis of some processes using reliability and statistical methods. In this context, probabilistic modelling of steels tenacity in nuclear plants tanks has been developed. The databases under consideration include hundreds of data indexed by temperature, so that, reliable probabilistic models have been obtained for the central part of the distribution.

However, in this reliability problem, the key point is to investigate the behaviour of the model in the distribution tail. In particular, we are mainly interested in studying the lowest tenacities when the temperature varies. We are currently investigating the opportunity to propose a postdoctoral position on this problem, supported by the CEA.