Stéphane Girard

National Institute for Research in Computer Science and Control | INRIA · MISTIS - Modelling and Inference of Complex and Structured Stochastic Systems Research Team

In this paper, we introduce a new risk measure, the so-called conditional tail moment. It is defined as the moment of order a ≥ 0 of the loss distribution above the upper α-quantile where α ∈ (0,1). Estimating the conditional tail moment permits us to estimate all risk measures based on conditional moments such as conditional tail expectation, cond...

Nonparametric regression quantiles obtained by inverting a kernel estimator of the conditional distribution of the response are long established in statistics. Attention has been, however, restricted to ordinary quantiles staying away from the tails of the conditional distribution. The purpose of this paper is to extend their asymptotic theory far...

This work presents a family of parsimonious Gaussian process models which allow to build, from a finite sample, a model-based classifier in an infinite dimensional space. The proposed parsimonious models are obtained by constraining the eigen-decomposition of the Gaussian processes modeling each class. This allows in particular to use non-linear ma...

In this paper, auto-associative models are proposed as candidates to the generalization of Principal Component Analysis. We show that these models are dedicated to the approximation of the dataset by a manifold. Here, the word "manifold" refers to the topology properties of the structure. The approximating manifold is built by a projection pursuit...

Sliced Inverse Regression (SIR) has been extensively used to reduce the dimension of the predictor space before performing regression. SIR is originally a model free method but it has been shown to actually correspond to the maximum likelihood of an inverse regression model with Gaussian errors. This intrinsic Gaussianity of standard SIR may explai...

We propose an extreme dimension reduction method extending the Extreme-PLS approach to the case where the covariate lies in a possibly infinite-dimensional Hilbert space. The ideas are partly borrowed from both Partial Least-Squares and Sliced Inverse Regression techniques. As such, the method relies on the projection of the covariate onto a subspa...

This work focuses on dimension-reduction techniques for modelling conditional extreme values. Specifically, we investigate the idea that extreme values of a response variable can be explained by nonlinear functions derived from linear projections of an input random vector. In this context, the estimation of projection directions is examined, as app...

This work focuses on dimension-reduction techniques for modelling conditional extreme values. Specifically, we investigate the idea that extreme values of a response variable can be explained by nonlinear functions derived from linear projections of an input random vector. In this context, the estimation of projection directions is examined, as app...

We propose new parametrizations for neural networks in order to estimate extreme quantiles in both non-conditional and conditional heavy-tailed settings. All proposed neural network estimators feature a bias correction based on an extension of the usual second-order condition to an arbitrary order. The convergence rate of the uniform error between...

Analysis of variance (ANOVA) is commonly employed to assess differences in the means of independent samples. However, it is unsuitable for evaluating differences in tail behaviour, especially when means do not exist or empirical estimation of moments is inconsistent due to heavy-tailed distributions. Here, we propose an ANOVA-like decomposition to...

The main goal of this report is to build a detector of all objects present in the surrounding of an autonomous vehicle on roads. We achieved this goal by means of a deep learning algorithm called YOLO (You Only Look Once) in the field of computer vision. The basic concepts in object detection as well as the core of the YOLO algorithm are recalled i...

Weissman extrapolation methodology for estimating extreme quantiles from heavy-tailed distributions is based on two estimators: an order statistic to estimate an intermediate quantile and an estimator of the tail-index. The common practice is to select the same intermediate sequence for both estimators. In this work, we show how an adapted choice o...

Combining extreme value theory with Bayesian methods offers several advantages, such as a quantification of uncertainty on parameter estimation or the ability to study irregular models that cannot be handled by frequentist statistics. However, it comes with many options that are left to the user concerning model building, computational algorithms,...

The classification of irregularly sampled Satellite image time-series (SITS) is investigated in this paper. A multivariate Gaussian process mixture model is proposed to address the irregular sampling, the multivariate nature of the time-series and the scalability to large data-sets. The spectral and temporal correlation is handled using a Kronecker...

We propose a new approach, called Extreme-PLS, for dimension reduction in conditional extreme values settings. The objective is to find linear combinations of covariates that best explain the extreme values of the response variable in a non-linear inverse regression model. The asymptotic normality of the Extreme-PLS estimator is established in the...

Expectiles induce a law-invariant risk measure that has recently gained popularity in actuarial and financial risk management applications. Unlike quantiles or the quantile-based Expected Shortfall, the expectile risk measure is coherent and elicitable. The estimation of extreme expectiles in the heavy-tailed framework, which is reasonable for extr...

Diagnosing convergence of Markov chain Monte Carlo is crucial and remains an essentially unsolved problem. Among the most popular methods, the potential scale reduction factor, commonly named $\hat{R}$, is an indicator that monitors the convergence of output chains to a target distribution, based on a comparison of the between- and within-variances...

We provide a large probability bound on the uniform approximation of fractional Brownian motion with Hurst parameter H, by a deep-feedforward ReLU neural network fed with a N-dimensional Gaussian vector, with bounds on the network design (number of hidden layers and total number of neurons). Essentially, up to log terms, achieving an uniform error...

The connection between Bayesian neural networks and Gaussian processes gained a lot of attention in the last few years, with the flagship result that hidden units converge to a Gaussian process limit when the layers width tends to infinity. Underpinning this result is the fact that hidden units become independent in the infinite-width limit. Our ai...

The connection between Bayesian neural networks and Gaussian processes gained a lot of attention in the last few years. Hidden units are proven to follow a Gaussian process limit when the layer width tends to infinity. Recent work has suggested that finite Bayesian neural networks may outperform their infinite counterparts because they adapt their...

Since its introduction in the early 90’s, the Sliced Inverse Regression (SIR) methodology has evolved adapting to increasingly complex data sets in contexts combining linear dimension reduction with non linear regression. The assumption of dependence of the response variable with respect to only a few linear combinations of the covariates makes it...

Quantiles are recognized tools for risk management and can be seen as minimizers of an L1-loss function, but do not define coherent risk measures in general. Expectiles, meanwhile, are minimizers of an L2-loss function and define coherent risk measures; they have started to be considered as good alternatives to quantiles in insurance and finance. Q...

Recent satellite missions have led to a huge amount of Earth observation data, most of them being freely available. In such a context, satellite image time series have been used to study land use and land cover information. However, optical time series, such as Sentinel-2 or Landsat ones, are provided with an irregular time sampling for different s...

Technical report

We propose a new measure of variability in the tail of a distribution by applying a Box–Cox transformation of parameter \(p \ge 0\) to the tail-Gini functional. It is shown that the so-called Box–Cox Tail Gini Variability measure is a valid variability measure whose condition of existence may be as weak as necessary thanks to the tuning parameter p...

Expectiles form a family of risk measures that have recently gained interest over the more common value-at-risk or return levels, primarily due to their capability to be determined by the probabilities of tail values and magnitudes of realisations at once. However, a prevalent and ongoing challenge of expectile inference is the problem of uncertain...

We consider a location-dispersion regression model for heavy-tailed distributions when the multidimensional covariate is deterministic. In a first step, nonparametric estimators of the regression and dispersion functions are introduced. This permits, in a second step, to derive an estimator of the conditional extreme-value index computed on the res...

The modeling of extreme events arises in many fields such as finance, insurance or environmental science. A recurrent statistical problem is then the estimation of extreme quantiles associated with a random variable $Y$ recorded simultaneously with a multidimensional covariate $x\in{\mathbb R}^d$, the goal being to describe how tail characteristic...

Expectiles and quantiles can both be defined as the solution of minimization problems. Contrary to quantiles though, expectiles are determined by tail expectations rather than tail probabilities, and define a coherent risk measure. For these two reasons in particular, expectiles have recently started to be considered as serious candidates to become...

Technical report

We propose the notion of sub‐Weibull distributions, which are characterised by tails lighter than (or equally light as) the right tail of a Weibull distribution. This novel class generalises the sub‐Gaussian and sub‐Exponential families to potentially heavier‐tailed distributions. Sub‐Weibull distributions are parameterized by a positive tail index...

The block maxima approach is one of the main methodologies in extreme value theory to obtain a suitable distribution to estimate the probability of large values. In this approach, the block size is usually selected in order to reflect the possible intrinsic periodicity of the studied phenomenon. The generalization of this approach to data from non-...

We acknowledge the priority on the introduction of the formula of t-lgHill estimator for the positive extreme value index. We provide a novel motivation for this estimator based on ecologically driven dynamical systems. Another motivation is given directly by applying the general t-Hill procedure to log-gamma distribution. We illustrate the good qu...

In this paper, we aim to assess and model the dependence structure between crop yields and prices, by using a copula approach. The study is conducted on a database of French farms by considering cereal and wine productions for years 2014 to 2016. We find that the dependence between prices and yields is relatively high and can be described with the...

Quantiles and expectiles can be interpreted as solutions of convex minimization problems. Unlike quantiles, expectiles are determined by tail expectations rather than tail probabilities , and define a coherent risk measure. For these reasons, among others, they have recently been the subject of renewed attention in actuarial and financial risk mana...

We investigate the asymptotic behavior of the (relative) extrapolation error associated with some estimators of extreme quantiles based on extreme-value theory. 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 te...

Risk measures of a financial position are, from an empirical point of view, mainly based on quantiles. Replacing quantiles with their least squares analogues, called expectiles, has recently received increasing attention. The novel expectile-based risk measures satisfy all coherence requirements. We revisit their extreme value estimation for heavy-...

We consider a location-dispersion regression model for heavy-tailed distributions when the multidimensional covariate is deterministic. In a first step, nonparametric estimators of the regression and dispersion functions are introduced. This permits, in a second step, to derive an estimator of the conditional extreme-value index computed on the res...

Expectiles define a least squares analogue of quantiles. They have been the focus of a substantial quantity of research in the context of actuarial and financial risk assessment over the last 10 years. Unlike quantiles, expectiles induce coherent risk measures and are calculated using tail expectations rather than merely tail probabilities ; contra...

We introduce a location-scale model for conditional heavy-tailed distributions when the covariate is deterministic. First, nonparametric estimators of the location and scale functions are introduced. Second, an estimator of the conditional extreme-value index is derived. The asymptotic properties of the estimators are established under mild assumpt...

The Conditional Tail Expectation is an indicator of tail behaviour that takes into account both the frequency and magnitude of a tail event. However, the asymptotic normality of its empirical estimator requires that the underlying distribution possess a finite variance; this can be a strong restriction in actuarial and financial applications. A val...

Quantiles are a fundamental concept in extreme-value theory. They can be obtained from a minimization framework using an absolute error loss criterion. The companion notion of expectiles, based on squared rather than absolute error loss minimization, has recently been receiving substantial attention from the fields of actuarial science, finance and...

We study a broad class of asymmetric copulas introduced by Liebscher (2008) as a combination of multiple - usually symmetric - copulas. The main thrust of the paper is to provide new theoretical properties including exact tail dependence expressions and stability properties. A subclass of Liebscher copulas obtained by combining Fr\'echet copulas is...

We study a broad class of asymmetric copulas introduced by Liebscher as a combination of multiple---usually symmetric---copulas. The main thrust of the paper is to provide new theoretical properties including exact tail dependence expressions and stability properties. A subclass of Liebscher copulas obtained by combining comonotonic copulas is st...

Expectiles and quantiles can both be defined as the solution of minimization problems. Contrary to quantiles though, expectiles are determined by tail expectations rather than tail probabilities , and define a coherent risk measure. For these two reasons in particular, expectiles have recently started to be considered as serious candidates to becom...

A new estimator for extreme quantiles is proposed under the log-generalized Weibull-tail model. This model relies on a new regular variation condition which, in some situations, permits to extrapolate further into the tails than the classical assumption in extreme-value theory. The asymptotic normality of the estimator is established and its finite...

Expectiles define a least squares analogue of quantiles. They are determined by tail expectations rather than tail probabilities. For this reason and many other theoretical and practical merits, expectiles have recently received a lot of attention, especially in actuarial and financial risk management. Their estimation, however, typically requires...

We investigate the properties of a new transformation of copulas based on the co-copula and an univariate function. It is shown that several families in the copula literature can be interpreted as particular outputs of this transformation. Symmetry, association, ordering and dependence properties of the resulting copula are established.

https://stat4astro2017.sciencesconf.org/

This paper is concerned with the estimation of a local measure of intrinsic dimensionality (ID) recently proposed by Houle. The local model can be regarded as an extension of Karger and Ruhl’s expansion dimension to a statistical setting in which the distribution of distances to a query point is modeled in terms of a continuous random variable. Thi...

The Regression Conditional Tail Moment (RCTM) is the risk measure defined as the moment of order b ≥ 0 of a loss distribution above the upper α-quantile where α ∈ (0, 1) and when a covariate information is available. The purpose of this work is first to establish the asymptotic properties of the RCTM in case of extreme losses, i.e when α → 0 is no...

Grasslands represent a significant source of biodiversity that is important to monitor over large extents. The Spectral Variation Hypothesis (SVH) assumes that the Spectral Heterogeneity (SH) measured from remote sensing data can be used as a proxy for species diversity. Here, we argue the hypothesis that the grassland’s species differ in their phe...

The class of quantiles lies at the heart of extreme-value theory and is one of the basic tools in risk management. The alternative family of expectiles is based on squared rather than absolute error loss minimization. It has recently been receiving a lot of attention in actuarial science, econometrics and statistical finance. Both quantiles and exp...

We use tail expectiles to estimate alternative measures to the Value at Risk (VaR) and Marginal Expected Shortfall (MES), two instruments of risk protection of utmost importance in actuarial science and statistical finance. The concept of expectiles is a least squares analogue of quantiles. Both are M-quantiles as the minimizers of an asymmetric co...

This paper deals with the classification of grasslands using high resolution satellite image time series. Grasslands considered in this work are semi-natural elements in fragmented landscapes, i.e., they are heterogeneous and small elements. The first contribution of this study is to account for grassland heterogeneity while working at the object l...

Score-Based Bayesian Network Structure Learning In Presence Of (Quasi-)Deterministic Relations

Our aim is to evaluate fundamental parameters from the analysis of the electromagnetic spectra of stars. We may use $10^3$-$10^5$ spectra; each spectrum being a vector with $10^2$-$10^4$ coordinates. We thus face the so-called "curse of dimensionality". We look for a method to reduce the size of this data-space, keeping only the most relevant infor...

We describe a novel method of heavy tails estimation based on transformed score (t-score). Based on a new score moment method we derive the t-Hill estimator, which estimates the extreme value index of a distribution function with regularly varying tail. t-Hill estimator is distribution sensitive, thus it differs in e.g. Pareto and log-gamma case. H...

We show how a dynamical system given by a t-score function for some class of monotonic data transformations generates consistent extreme value estimators. The variation of their values increases the uncertainty of proper assessment of climate change. Two important examples illustrate the methodology: mass balance measurements on Guanaco glacier, Ch...