Section: New Results

Semi and non-parametric methods

Conditional extremal events

Participant : Stéphane Girard.

Joint work with: L. Gardes (Univ. Strasbourg), A. Daouia (Univ. Toulouse I and Univ. Catholique de Louvain), J. Elmethni (Univ. Paris 5) and S. Louhichi (Univ. Grenoble 1)

The goal of the PhD thesis of Alexandre Lekina was to contribute to the development of theoretical and algorithmic models to tackle conditional extreme value analysis, ie the situation where some covariate information $X$ is recorded simultaneously with a quantity of interest $Y$. In such a case, the tail heaviness of $Y$ depends on $X$, and thus the tail index as well as the extreme quantiles are also functions of the covariate. We combine nonparametric smoothing techniques  [71] with extreme-value methods in order to obtain efficient estimators of the conditional tail index and conditional extreme quantiles. The strong consistency of such estimator is established in [53] . When the covariate is functional and random (random design) we focus on kernel methods [58] .

Conditional extremes are studied in climatology where one is interested in how climate change over years might affect extreme temperatures or rainfalls. In this case, the covariate is univariate (time). Bivariate examples include the study of extreme rainfalls as a function of the geographical location. The application part of the study is joint work with the LTHE (Laboratoire d'étude des Transferts en Hydrologie et Environnement) located in Grenoble.

Estimation of extreme risk measures

Participant : Stéphane Girard.

Joint work with: E. Deme (Univ. Gaston-Berger, Sénégal, J. Elmethni (Univ. Paris 5), L. Gardes and A. Guillou (Univ. Strasbourg)

One of the most popular risk measures is the Value-at-Risk (VaR) introduced in the 1990's. In statistical terms, the VaR at level $\alpha \in (0,1)$ corresponds to the upper $\alpha$-quantile of the loss distribution. The Value-at-Risk however suffers from several weaknesses. First, it provides us only with a pointwise information: VaR($\alpha$) does not take into consideration what the loss will be beyond this quantile. Second, random loss variables with light-tailed distributions or heavy-tailed distributions may have the same Value-at-Risk . Finally, Value-at-Risk is not a coherent risk measure since it is not subadditive in general. A coherent alternative risk measure is the Conditional Tail Expectation (CTE), also known as Tail-Value-at-Risk, Tail Conditional Expectation or Expected Shortfall in case of a continuous loss distribution. The CTE is defined as the expected loss given that the loss lies above the upper $\alpha$-quantile of the loss distribution. This risk measure thus takes into account the whole information contained in the upper tail of the distribution. It is frequently encountered in financial investment or in the insurance industry. In [52] , we have established the asymptotic properties of the CTE estimator in case of extreme losses, i.e. when $\alpha \to 0$ as the sample size increases. We have exhibited the asymptotic bias of this estimator, and proposed a bias correction based on extreme-value techniques. In [20] , we study the situation where some covariate information is available. We thus has to deal with conditional extremes (see paragraph  6.5.1 ). We also proposed a new risk measure (called the Conditional Tail Moment) which encompasses various risk measures, such as the CTE, as particular cases.

Multivariate extremal events

Participants : Stéphane Girard, Gildas Mazo, Florence Forbes.

Joint work with: C. Amblard (TimB in TIMC laboratory, Univ. Grenoble I), L. Gardes (Univ. Strasbourg) and L. Menneteau (Univ. Montpellier II)

Copulas are a useful tool to model multivariate distributions  [75] . At first, we developed an extension of some particular copulas [1] . It followed a new class of bivariate copulas defined on matrices [55] and some analogies have been shown between matrix and copula properties.

However, while there exist various families of bivariate copulas, much fewer has been done when the dimension is higher. To this aim an interesting class of copulas based on products of transformed copulas has been proposed in the literature. The use of this class for practical high dimensional problems remains challenging. Constraints on the parameters and the product form render inference, and in particular the likelihood computation, difficult. We proposed a new class of high dimensional copulas based on a product of transformed bivariate copulas [64] . No constraints on the parameters refrain the applicability of the proposed class which is well suited for applications in high dimension. Furthermore the analytic forms of the copulas within this class allow to associate a natural graphical structure which helps to visualize the dependencies and to compute the likelihood efficiently even in high dimension. The extreme properties of the copulas are also derived and an R package has been developed.

As an alternative, we also proposed a new class of copulas constructed by introducing a latent factor. Conditional independence with respect to this factor and the use of a nonparametric class of bivariate copulas lead to interesting properties like explicitness, flexibility and parsimony. In particular, various tail behaviours are exhibited, making possible the modeling of various extreme situations [42] . A pairwise moment-based inference procedure has also been proposed and the asymptotic normality of the corresponding estimator has been established [66] .

In collaboration with L. Gardes, we investigate the estimation of the tail copula which is widely used to describe the amount of extremal dependence of a multivariate distribution. In some situations such as risk management, the dependence structure can be linked with some covariate. The tail copula thus depends on this covariate and is referred to as the conditional tail copula. The aim of our work is to propose a nonparametric estimator of the conditional tail copula and to establish its asymptotic normality [57] .

Level sets estimation

Participant : Stéphane Girard.

Joint work with: A. Guillou and L. Gardes (Univ. Strasbourg), A. Nazin (Univ. Moscou), G. Stupfler (Univ. Aix-Marseille) and A. Daouia (Univ. Toulouse I and Univ. Catholique de Louvain)

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 proposed estimators based on projection as well as on kernel regression methods applied on the extreme values set, for particular set of points [10] . We also investigate the asymptotic properties of existing estimators when used in extreme situations. For instance, we have established in collaboration with G. Stupfler that the so-called geometric quantiles have very counter-intuitive properties in such situations [63] , [62] and thus should not be used to detect outliers. These resuls are submitted for publication.

In collaboration with A. Daouia, we investigate the application of such methods in econometrics [16] : A new characterization of partial boundaries of a free disposal multivariate support is introduced by making use of large quantiles of a simple transformation of the underlying multivariate distribution. Pointwise empirical and smoothed estimators of the full and partial support curves are built as extreme sample and smoothed quantiles. The extreme-value theory holds then automatically for the empirical frontiers and we show that some fundamental properties of extreme order statistics carry over to Nadaraya's estimates of upper quantile-based frontiers.

In collaboration with A. Nazin, we define new estimators of the frontier function based on linear programming methods. The frontier is defined as the solution of a linear optimization problem under inequality constraints. The estimator is shown to be strongly consistent with respect to the $L_1$ norm and we establish that it reaches the optimal minimax rate of convergence [58] .

In collaboration with G. Stupfler and A. Guillou, new estimators of the boundary are introduced. The regression is performed on the whole set of points, the selection of the “highest” points being automatically performed by the introduction of high order moments [22] .

Retrieval of Mars surface physical properties from OMEGA hyperspectral images.

Participants : Stéphane Girard, Alessandro Chiancone.

Joint work with: S. Douté from Laboratoire de Planétologie de Grenoble, J. Chanussot (Gipsa-lab and Grenoble-INP) and J. Saracco (Univ. Bordeaux).

Visible and near infrared imaging spectroscopy is one of the key techniques to detect, to map and to characterize mineral and volatile (eg. water-ice) species existing at the surface of planets. Indeed the chemical composition, granularity, texture, physical state, etc. of the materials determine the existence and morphology of the absorption bands. The resulting spectra contain therefore very useful information. Current imaging spectrometers provide data organized as three dimensional hyperspectral images: two spatial dimensions and one spectral dimension. Our goal is to estimate the functional relationship $F$ between some observed spectra and some physical parameters. To this end, a database of synthetic spectra is generated by a physical radiative transfer model and used to estimate $F$. The high dimension of spectra is reduced by Gaussian regularized sliced inverse regression (GRSIR) to overcome the curse of dimensionality and consequently the sensitivity of the inversion to noise (ill-conditioned problems) [15] . We have also defined an adaptive version of the method which is able to deal with block-wise evolving data streams [13] .

In his PhD thesis work, Alessandro Chiancone studies the extension of the SIR method to different sub-populations. The idea is to assume that the dimension reduction subspace may not be the same for different clusters of the data [46] . He also published a paper on a previous work in the field of hierarchical segmentation of images [14] .