arXiv subjects

Anna Kiriliouk

Publications and source records attributed to Anna Kiriliouk.

15 recordsLinked to original sources

arXiv2026

A sub-asymptotic model for bivariate threshold exceedances

Extreme value theory offers a statistical framework for quantifying the risk of rare events, with the generalized Pareto (GP) distribution providing the canonical limit model for univariate threshold exceedances. In many applications, however, extremes are intrinsically multivariate, requiring models that capture both marginal tail behaviours and joint extremal dependencies. Under asymptotic dependence, the multivariate GP distribution represents a suitable modelling family, but when asymptotic independence arises, sub-asymptotic models are needed. In this work, we propose and study a flexible sub-asymptotic parametric class to model bivariate threshold exceedances. Our new model accommodates a broad range of tail dependence behaviours and contains the standardised multivariate GP distribution as a limiting case while retaining margins that converge to univariate GP tails. Our formulation allows extremal dependence to evolve naturally with the marginal parameters on the original data scale, facilitating direct computation and interpretation of failure probabilities. Model inference is done via a likelihood-free neural Bayes estimation approach, with tailored prior specifications. An extensive simulation study and an application to Belgian rainfall extremes illustrate the estimation framework and the flexibility of the model.

stat.ME↗

arXiv2025

A penalized least squares estimator for extreme-value mixture models

Estimating the parameters of max-stable parametric models poses significant challenges, particularly when some parameters lie on the boundary of the parameter space. This situation arises when a subset of variables exhibits extreme values simultaneously, while the remaining variables do not -- a phenomenon commonly referred to as an extreme direction. A novel estimator is proposed for the parameters of a general parametric mixture model, incorporating a threshold exceedances approach based on a pseudo-norm penalization. The latter plays a crucial role in accurately identifying parameters at the boundary of the parameter space. Additionally, the estimator comes with a data-driven algorithm to detect groups of variables corresponding to extreme directions. The performance of the estimator is assessed in terms of both parameter estimation and the identification of extreme directions through extensive simulation studies. Finally, the method is applied to two real-world datasets: discharge measurements at stations along the Danube river, and financial portfolio losses from stocks listed on the NYSE, AMEX, and NASDAQ. In both applications, the sets of variables that can become large simultaneously are identified.

stat.ME↗

arXiv2024

Tail Risk Analysis for Financial Time Series

This book chapter illustrates how to apply extreme value statistics to financial time series data. Such data often exhibits strong serial dependence, which complicates assessment of tail risks. We discuss the two main approches to tail risk estimation, unconditional and conditional quantile forecasting. We use the S&P 500 index as a case study to assess serial (extremal) dependence, perform an unconditional and conditional risk analysis, and apply backtesting methods. Additionally, the chapter explores the impact of serial dependence on multivariate tail dependence.

q-fin.RM↗

arXiv2024

Multivariate generalized Pareto distributions along extreme directions

When modeling a vector of risk variables, extreme scenarios are often of special interest. The peaks-over-thresholds method hinges on the notion that, asymptotically, the excesses over a vector of high thresholds follow a multivariate generalized Pareto distribution. However, existing literature has primarily concentrated on the setting when all risk variables are always large simultaneously. In reality, this assumption is often not met, especially in high dimensions. In response to this limitation, we study scenarios where distinct groups of risk variables may exhibit joint extremes while others do not. These discernible groups are derived from the angular measure inherent in the corresponding max-stable distribution, whence the term extreme direction. We explore such extreme directions within the framework of multivariate generalized Pareto distributions, with a focus on their probability density functions in relation to an appropriate dominating measure. Furthermore, we provide a stochastic construction that allows any prespecified set of risk groups to constitute the distribution's extreme directions. This construction takes the form of a smoothed max-linear model and accommodates the full spectrum of conceivable max-stable dependence structures. Additionally, we introduce a generic simulation algorithm tailored for multivariate generalized Pareto distributions, offering specific implementations for extensions of the logistic and Hüsler-Reiss families capable of carrying arbitrary extreme directions.

math.ST↗

arXiv2024

X-Vine Models for Multivariate Extremes

Regular vine sequences permit the organisation of variables in a random vector along a sequence of trees. Regular vine models have become greatly popular in dependence modelling as a way to combine arbitrary bivariate copulas into higher-dimensional ones, offering flexibility, parsimony, and tractability. In this project, we use regular vine structures to decompose and construct the exponent measure density of a multivariate extreme value distribution, or, equivalently, the tail copula density. Although these densities pose theoretical challenges due to their infinite mass, their homogeneity property offers simplifications. The theory sheds new light on existing parametric families and facilitates the construction of new ones, called X-vines. Computations proceed via recursive formulas in terms of bivariate model components. We develop simulation algorithms for X-vine multivariate Pareto distributions as well as methods for parameter estimation and model selection on the basis of threshold exceedances. The methods are illustrated by Monte Carlo experiments and a case study on US flight delay data.

stat.ME↗

arXiv2022

Estimating probabilities of multivariate failure sets based on pairwise tail dependence coefficients

Estimating probabilities of extreme events involving multiple risk factors is a critical challenge in fields such as finance and climate science. This paper proposes a parametric approach to estimate the probability that a multivariate random vector falls into an extreme failure set, based on the information in the tail pairwise dependence matrix (TPDM) only. The TPDM provides a summary of tail dependence for all pairs of components of the random vector. We propose an efficient algorithm to obtain approximate completely positive decompositions of the TPDM, enabling the construction of a max-linear model whose TPDM approximates that of the original random vector. We also provide conditions under which the approximation turns out to be exact. Based on the decompositions, we can construct max-linear random vectors to estimate failure probabilities, exploiting their computational simplicity. We apply the proposed method to estimate probabilities of extreme events for real-world datasets, including industry portfolio returns and maximal wind speeds, demonstrating its practical utility for risk assessment.

stat.ME↗

arXiv2020

Climate extreme event attribution using multivariate peaks-over-thresholds modeling and counterfactual theory

Numerical climate models are complex and combine a large number of physical processes. They are key tools in quantifying the relative contribution of potential anthropogenic causes (e.g., the current increase in greenhouse gases) on high impact atmospheric variables like heavy rainfall. These so-called climate extreme event attribution problems are particularly challenging in a multivariate context, that is, when the atmospheric variables are measured on a possibly high-dimensional grid. In this paper, we leverage two statistical theories to assess causality in the context of multivariate extreme event attribution. As we consider an event to be extreme when at least one of the components of the vector of interest is large, extreme-value theory justifies, in an asymptotical sense, a multivariate generalized Pareto distribution to model joint extremes. Under this class of distributions, we derive and study probabilities of necessary and sufficient causation as defined by the counterfactual theory of Pearl. To increase causal evidence, we propose a dimension reduction strategy based on the optimal linear projection that maximizes such causation probabilities. Our approach is tested on simulated examples and applied to weekly winter maxima precipitation outputs of the French CNRM from the recent CMIP6 experiment.

stat.AP↗

arXiv2020

Resampling Procedures with Empirical Beta Copulas

The empirical beta copula is a simple but effective smoother of the empirical copula. Because it is a genuine copula, from which, moreover, it is particularly easy to sample, it is reasonable to expect that resampling procedures based on the empirical beta copula are expedient and accurate. In this paper, after reviewing the literature on some bootstrap approximations for the empirical copula process, we first show the asymptotic equivalence of several bootstrapped processes related to the empirical copula and empirical beta copula. Then we investigate the finite-sample properties of resampling schemes based on the empirical (beta) copula by Monte Carlo simulation. More specifically, we consider interval estimation for some functionals such as rank correlation coefficients and dependence parameters of several well-known families of copulas, constructing confidence intervals by several methods and comparing their accuracy and efficiency. We also compute the actual size and power of symmetry tests based on several resampling schemes for the empirical copula and empirical beta copula.

math.ST↗

arXiv2018

Hypothesis testing for tail dependence parameters on the boundary of the parameter space

Modelling multivariate tail dependence is one of the key challenges in extreme-value theory. Multivariate extremes are usually characterized using parametric models, some of which have simpler submodels at the boundary of their parameter space. Hypothesis tests are proposed for tail dependence parameters that, under the null hypothesis, are on the boundary of the alternative hypothesis. The asymptotic distribution of the weighted least squares estimator (Einmahl, Kiriliouk and Segers, Extremes 21, pages 205-233, 2018) is given when the true parameter vector is on the boundary of the parameter space, and two test statistics are proposed. The performance of these test statistics is evaluated for the Brown-Resnick model and the max-linear model. In particular, simulations show that it is possible to recover the optimal number of factors for a max-linear model. Finally, the methods are applied to characterize the dependence structure of two major stock market indices, the DAX and the CAC40.

stat.ME↗

arXiv2018

Peaks over thresholds modelling with multivariate generalized Pareto distributions

When assessing the impact of extreme events, it is often not just a single component, but the combined behaviour of several components which is important. Statistical modelling using multivariate generalized Pareto (GP) distributions constitutes the multivariate analogue of univariate peaks over thresholds modelling, which is widely used in finance and engineering. We develop general methods for construction of multivariate GP distributions and use them to create a variety of new statistical models. A censored likelihood procedure is proposed to make inference on these models, together with a threshold selection procedure, goodness-of-fit diagnostics, and a computationally tractable strategy for model selection. The models are fitted to returns of stock prices of four UK-based banks and to rainfall data in the context of landslide risk estimation. Supplementary materials and codes are available online.

stat.ME↗

arXiv2017

An estimator of the stable tail dependence function based on the empirical beta copula

The replacement of indicator functions by integrated beta kernels in the definition of the empirical stable tail dependence function is shown to produce a smoothed version of the latter estimator with the same asymptotic distribution but superior finite-sample performance. The link of the new estimator with the empirical beta copula enables a simple but effective resampling scheme.

stat.ME↗

arXiv2016

A continuous updating weighted least squares estimator of tail dependence in high dimensions

Likelihood-based procedures are a common way to estimate tail dependence parameters. They are not applicable, however, in non-differentiable models such as those arising from recent max-linear structural equation models. Moreover, they can be hard to compute in higher dimensions. An adaptive weighted least-squares procedure matching nonparametric estimates of the stable tail dependence function with the corresponding values of a parametrically specified proposal yields a novel minimum-distance estimator. The estimator is easy to calculate and applies to a wide range of sampling schemes and tail dependence models. In large samples, it is asymptotically normal with an explicit and estimable covariance matrix. The minimum distance obtained forms the basis of a goodness-of-fit statistic whose asymptotic distribution is chi-square. Extensive Monte Carlo simulations confirm the excellent finite-sample performance of the estimator and demonstrate that it is a strong competitor to currently available methods. The estimator is then applied to disentangle sources of tail dependence in European stock markets.

stat.ME↗

arXiv2015

An M-estimator of spatial tail dependence

Tail dependence models for distributions attracted to a max-stable law are fitted using observations above a high threshold. To cope with spatial, high-dimensional data, a rank-based M-estimator is proposed relying on bivariate margins only. A data-driven weight matrix is used to minimize the asymptotic variance. Empirical process arguments show that the estimator is consistent and asymptotically normal. Its finite-sample performance is assessed in simulation experiments involving popular max-stable processes perturbed with additive noise. An analysis of wind speed data from the Netherlands illustrates the method.

stat.ME↗

arXiv2014

Max-factor individual risk models with application to credit portfolios

Individual risk models need to capture possible correlations as failing to do so typically results in an underestimation of extreme quantiles of the aggregate loss. Such dependence modelling is particularly important for managing credit risk, for instance, where joint defaults are a major cause of concern. Often, the dependence between the individual loss occurrence indicators is driven by a small number of unobservable factors. Conditional loss probabilities are then expressed as monotone functions of linear combinations of these hidden factors. However, combining the factors in a linear way allows for some compensation between them. Such diversification effects are not always desirable and this is why the present work proposes a new model replacing linear combinations with maxima. These max-factor models give more insight into which of the factors is dominant.

stat.ME↗

arXiv2014

Nonparametric estimation of extremal dependence

There is an increasing interest to understand the dependence structure of a random vector not only in the center of its distribution but also in the tails. Extreme-value theory tackles the problem of modelling the joint tail of a multivariate distribution by modelling the marginal distributions and the dependence structure separately. For estimating dependence at high levels, the stable tail dependence function and the spectral measure are particularly convenient. These objects also lie at the basis of nonparametric techniques for modelling the dependence among extremes in the max-domain of attraction setting. In case of asymptotic independence, this setting is inadequate, and more refined tail dependence coefficients exist, serving, among others, to discriminate between asymptotic dependence and independence. Throughout, the methods are illustrated on financial data.

stat.ME↗