1.1. The PhD thesis of Clément Albert (Mistis) has been defended. This thesis takes place in the (univariate) extreme risk analysis framework. It focuses on extreme quantile estimation which is a two step approach. First, it consists in proposing an extreme value based quantile approximation. Then, estimators of the unknown quantities are plugged in the previous approximation leading to an extreme quantile estimator. The first contribution of the thesis is the study of the extrapolation error, which is the error due to the extreme value based approximation of the true quantile. It is shown that the extrapolation error can be interpreted as the remainder of a first order Taylor expansion. Necessary and sufficient conditions are then provided such that this error tends to zero as the sample size increases. First order equivalents of the extrapolation error are derived and their accuracy is illustrated numerically (C. Albert et al, 2020a). The second contribution is the proposition of a new extreme quantile estimator. The problem is addressed in the framework of the so-called "log-Generalized Weibull tail limit" model. Based on this model, estimators of the parameters are proposed. Then, a new estimator of extreme quantiles is derived from the latter. Its asymptotic normality is established and its behavior in practice is illustrated on both real and simulated data (C. Albert et al, 2020b). We also proposed a new estimator of the tail-index of a distribution based on extreme Lp-quantiles. The asymptotic normality of such an estimator has been established, and, in doing so, we extended very recent results on extreme expectile and Lp-quantile estimation. We provided a discussion of the choice of p in practice, as well as a methodology for reducing the bias of our estimator. Its finite-sample performance is evaluated on simulated data and on a set of real hydrological data. The working paper is submitted for publication.
C. Albert, A. Dutfoy and S. Girard. Asymptotic behavior of the extrapolation error associated with the estimation of extreme quantiles, Extremes, 23, 349--380, 2020a [Associated technical report: pdf].
C. Albert, A. Dutfoy, L. Gardes and S. Girard. An extreme quantile estimator for the log-generalized Weibull-tail model, Econometrics and Statistics, 13, 137--174, 2020b [Associated technical report: pdf].
S. Girard, G. Stupfler and A. Usseglio-Carleve. An Lp-quantile methodology for tail index estimation, 2020, [pdf].
1.2. Recall that our aim is to propose new estimators of extreme risk measures able to deal with covariates. First, we investigated the estimation of extreme expectiles in a regression context. We exploited the fact that the expectiles of a distribution F are in fact the quantiles of another distribution E explicitly linked to F, in order to construct nonparametric kernel estimators of extreme conditional expectiles. We analyzed the asymptotic properties of our estimators in the context of conditional heavy-tailed distributions. Applications to simulated data and real insurance data are provided. The working paper is submitted for publication. Second, the PhD thesis of Aboubacrène Ag Ahmad has be defended. We defined a location-scale model for heavy-tailed distributions where the covariate is deterministic. We first address the non-parametric estimation of the location and scale functions and derive an estimator of the conditional extreme-value index (A. A. Ahmad et al, 2019). Then, new estimators of the extreme conditional quantiles are introduced. The asymptotic properties of the estimators are established. Such estimators are called semi-parametric, since they involve a non-parametric part (through the location and scale functions) as well as a parametric one (through the extreme-value index). The main interest of such models is that they inherit both flexibility from the non-parametric part and robustness from the parametric one. The working paper is submitted for publication.
S. Girard, G. Stupfler and A. Usseglio-Carleve. Nonparametric extreme conditional expectile estimation, 2020, [pdf].
A. A. Ahmad, E. Deme, A. Diop and S. Girard. Estimation of the tail-index in a conditional location-scale family of heavy-tailed distributions, Dependence Modeling, 7, 394--417, 2019. [associated technical report: pdf].
A. A. Ahmad, E. Deme, A. Diop, S. Girard and A. Usseglio-Carleve, Estimation of extreme quantiles from heavy-tailed distributions in a location-dispersion regression model, 2020, [pdf].
1.3. The estimation of the tail-index has also been investigated in the challenging situation where the covariate is a random field.
A. Bassene, S. Dabo-Niang, A. Diop and B. Thiam. Conditional tail index estimation for alpha-mixing random fields, 2019.
S. Dabo-Niang, A. Diop and B. Tchamiè. Conditional tail index and quantile estimation for beta-mixing random fields, 2019.
2.1. In the framework of the PhD thesis of Alexandre Constantin, a new statistical approach using Gaussian processes has been proposed to classify irregularly sampled signals without temporal rescaling. The model offers a theoretical framework to impute missing values. First experiments on simulated data show promising resultsboth in terms of classification and imputation accuracy. Good robustness properties with respect to the modelling assumptions are also observed. The method is currently tested on real satellite data. First results have been presented at two conferences.
A. Constantin, M. Fauvel, S. Girard and S. Iovleff. Classification de signaux multidimensionnels irrégulièrement échantillonnés, Vingt-septième colloque GRETSI, Lille, 2019. [pdf]
2.2. Unlike classical clustering, which groups similar objects from a single collection of objects, co-clustering or bi-clustering aims at simultaneously grouping objects from two disjoint sets, thus revealing interactions between elements of two sets. In recent years, co-clustering has been increasingly used in many areas ranging from information retrieval, data mining, computer vision, biology, ... We developed a new co-clustering method using covariates with application to genomic data. The theory and the implementation have been completed and the method has been tested on genomic data provided by Institut Pasteur de Dakar. The working paper is submitted for publication. SIMERG2E is contributing to the Rmixmod software (R interface of MIXMOD software for supervised, unsupervised and semi-supervised classification with MIXture MODelling) which is freely available on the CRAN (Comprehensive R Archive Network) and distributed under a GPL-3 licence. Some team members are also involved in the blockcluster, MixAll and rtkore packages. Some of these tools were developed to address Task 2.2.
S. Iovleff, S. Sylla and C. Loucoubar. Co-clustering binary data using covariates, 2018. [pdf].
S. Iovleff and S. Sylla. "Blockcluster" and "Simerge": Two R packages for Latent Block Models and Latent Block Models with co-variables implemented in C++, Mixture models: Theory and applications, Paris, 2018. [pdf].