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Section: New Results

Semi and non-parametric methods

Conditional extremal events

Participant : Stéphane Girard.

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

The goal of the PhD theses of Alexandre Lekina and Jonathan El Methni 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  [63] with extreme-value methods in order to obtain efficient estimators of the conditional tail index and conditional extreme quantiles. The strong consistency of such an estimator is established in [53] . When the covariate is functional and random (random design) we focus on kernel methods [23] .

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 [20] and the “département Génie urbain” of “Université Paris-Est Marne-la-vallée” [11] .

Estimation of extreme risk measures

Participant : Stéphane Girard.

Joint work with: A. Daouia (Univ. Toulouse), E. Deme (Univ. Gaston-Berger, Sénégal), A. Guillou (Univ. Strasbourg) and G. Stupfler (Univ. Aix-Marseille).

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 first 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 [51] , 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. A second possible coherent alternative risk measure is based on expectiles [59] . Compared to quantiles, the family of expectiles is based on squared rather than absolute error loss minimization. The flexibility and virtues of these least squares analogues of quantiles are now well established in actuarial science, econometrics and statistical finance. Both quantiles and expectiles were embedded in the more general class of M-quantiles as the minimizers of a generic asymmetric convex loss function. It has been proved very recently that the only M-quantiles that are coherent risk measures are the expectiles.

Multivariate extremal events

Participants : Stéphane Girard, Florence Forbes.

Joint work with: F. Durante (Univ. Bolzen-Bolzano, Italy) L. Gardes (Univ. Strasbourg) and G. Mazo (Univ. Catholique de Louvain, Belgique).

Copulas are a useful tool to model multivariate distributions  [67] .

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 [26] . 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 [19] , [27] , [52] . A pairwise moment-based inference procedure has also been proposed and the asymptotic normality of the corresponding estimator has been established [28] .

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 [22] .

Level sets estimation

Participant : Stéphane Girard.

Joint work with: G. Stupfler (Univ. Aix-Marseille)

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 [24] , [25] and thus should not be used to detect outliers.

Retrieval of Mars surface physical properties from OMEGA hyperspectral images.

Participants : Stéphane Girard, Alessandro Chiancone.

Joint work with: J. Chanussot (Gipsa-lab and Grenoble-INP).

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.

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 [14] .

Robust Sliced Inverse Regression.

Participants : Stéphane Girard, Alessandro Chiancone, Florence Forbes.

Sliced Inverse Regression (SIR) has been extensively used to reduce the dimension of the predictor space before performing regression. Recently it has been shown that this techniques is, not surprisingly, sensitive to noise. Different approaches has been proposed to robustify SIR, in this work, we start considering an inverse problem proposed by R.D. Cook and we show that the framework can be extended to take into account a non-Gaussian noise. Generalized Student distribution are considered and all parameters are estimated via EM algorithm. The algorithm is outlined and tested comparing the results with different approaches on simulated data. Results on a real dataset shows the interest of this technique in presence of outliers.

Robust Locally linear mapping with mixtures of Student distributions

Participants : Florence Forbes, Emeline Perthame, Brice Olivier, Leo Nicoletti.

The standard GLLiM model [17] for high dimensional regression assumes Gaussian noise models and is in its unconstrained version equivalent to a joint GMM. The fact that response and independent variables $(X,Y)$ are jointly a mixture of Gaussian distribution is the key for all derivations in the model. In this work, we show that similar developments are possible based on a joint Student Mixture model, joint SMM. It follows a new model referred to as SLLiM for Student Locally linear mapping for which we investigate the robustness to outlying data in a high dimensional regression context.