Recent advances in extreme value theory

Thursday, April 20th, 6:00 PT / 9:00 ET / 15:00 CET

Extreme value theory is concerned with the accurate statistical assessment of the risk of rare events. Such extreme events have small occurrence probabilities but often entail large economic and ecological costs in addition to their potential for severe impacts on human health. There is active research to model and infer complex extremal dependence structures in multivariate data, with connections to high-dimensional statistics, machine learning and other fields. Applications include the analysis of financial crises, prediction of flood risk and the modeling of record-shattering climate extremes.

In the webinar, selected statisticians will present their recent works and elaborate on different aspects of this topic.

When & Where:

• Thursday, April 20th, 6:00 PT / 9:00 ET / 15:00 CET
• Online, via Zoom. The registration form is available here.

Speakers:

• Stefano Rizzelli, Department of Statistical Sciences, Catholic University of Milan

Title: Mathematical foundations of empirical Bayesian analysis of maxima

Abstract: Predicting future observations is the central goal of several statistical applications concerning extreme value data. Under mild assumptions, extreme-value theory justifies modelling linearly normalized sample maxima by max-stable distributions, which provide asymptotic approximations to the actual data generating mechanism. The Bayesian paradigm offers a direct approach to forecasting and uncertainty quantification. Various proposals for Bayesian inferential procedures have been formulated in recent years, though they typically disregard the model convergence bias inherent in the use of max-stable distributions, incorporating no information on norming sequences in prior specifications for scale and location parameters. We propose an empirical Bayes approach which suitably addresses this point via data-dependent priors. We illustrate the resulting asymptotic posterior concentration properties and pinpoint their implications for estimation and prediction of future observations. This talk is based on a joint work with Simone A. Padoan.

• Nicola Gnecco, Department of Mathematical Sciences, University of Copenhagen

Title: Extremal Random Forests

Abstract: Classical methods for quantile regression fail in cases where the quantile of interest is extreme and only few or no training data points exceed it. Asymptotic results from extreme value theory can be used to extrapolate beyond the range of the data, and several approaches exist that use linear regression, kernel methods or generalized additive models. Most of these methods break down if the predictor space has more than a few dimensions or if the regression function of extreme quantiles is complex. We propose a method for extreme quantile regression that combines the flexibility of random forests with the theory of extrapolation. Our extremal random forest (ERF) estimates the parameters of a generalized Pareto distribution, conditional on the predictor vector, by maximizing a local likelihood with weights extracted from a quantile random forest. Under certain assumptions, we show consistency of the estimated parameters. Furthermore, we penalize the shape parameter in this likelihood to regularize its variability in the predictor space. Simulation studies show that our ERF outperforms both classical quantile regression methods and existing regression approaches from extreme value theory. We apply our methodology to extreme quantile prediction for U.S. wage data.

• Jonathan Koh, Oeschger Centre for Climate Change Research, University of Bern

Title: Predicting risks of temperature extremes using large-scale circulation patterns with r–Pareto processes

Abstract: Many severe weather patterns in the mid-latitudes have been found to be connected to a particular atmospheric pattern known as blocking. This pattern obstructs the prevailing westerly large-scale atmospheric flow, changing flow anomalies in the vicinity of the blocking system to sustain weather conditions in the immediate region of its occurrence. Blockings’ presence and characteristics are thus important for the development of temperature extremes, which are rarely isolated in space, so one must not just account for their occurrence probabilities and intensities but also their spatial dependencies when assessing their associated risk. Here we propose a methodology that does so by combining tools from the spatial extremes and machine learning literature, to incorporate 500hPa geopotential (Z500) anomalies over the North Atlantic and European region as covariates to predict surface temperature extremes. This involves fitting Generalized r-Pareto processes with appropriate risk functionals to high-impact positive and negative temperature anomaly events across central Europe from 1979–2020, using loss functions motivated by extreme-value theory in a boosting algorithm. We check by simulation that the model parameters are identifiable and can be estimated adequately. We find which circulation patterns in the Euro-Atlantic sector are most important in determining the characteristics of these extremes, and show how they affect it.

Discussant: Sebastian Engelke, Research Center for Statistics, University of Geneva

The webinar is part of YoungStatS project of the Young Statisticians Europe initiative (FENStatS) supported by the Bernoulli Society for Mathematical Statistics and Probability and the Institute of Mathematical Statistics (IMS).