arXiv subjects

Simone A. Padoan

Publications and source records attributed to Simone A. Padoan.

At least 19 recordsLinked to original sources

arXiv2025

Accurate Bayesian inference for tail risk extrapolation in time series

Accurately quantifying tail risks-rare but high-impact events such as financial crashes or extreme weather-is a central challenge in risk management, with serially dependent data. We develop a Bayesian framework based on the Generalized Pareto (GP) distribution for modeling threshold exceedances, providing posterior distributions for the GP parameters and tail quantiles in time series. Two cases are considered: extrapolation of tail quantiles for the stationary marginal distribution under beta-mixing dependence, and dynamic, past-conditional tail quantiles in heteroscedastic regression models. The proposal yields asymptotically honest credible regions, whose coverage probabilities converge to their nominal levels. We establish the asymptotic theory for the Bayesian procedure, deriving conditions on the prior distributions under which the posterior satisfies key asymptotic properties. To achieve this, we first develop a likelihood theory under serial dependence, providing local and global bounds for the empirical log-likelihood process of the misspecified GP model and deriving corresponding asymptotic properties of the Maximum Likelihood Estimator (MLE). Simulations demonstrate that our Bayesian credible regions outperform naive Bayesian and MLE-based confidence regions across several standard time series models, including ARMA, GARCH, and Markovian copula models. Two real-data applications-to U.S. interest rates and Swiss electricity demand-highlight the relevance of the proposed methodology.

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arXiv2025

Asymptotic theory for the likelihood-based block maxima method in time series

This paper develops a rigorous asymptotic framework for likelihood-based inference in the Block Maxima (BM) method for stationary time series. While Bayesian inference under the BM approach has been widely studied in the independence setting, no asymptotic theory currently exists for time series. Further results are needed to establish that BM method can be applied with the kind of dependent time series models relevant to applied fields. To address this gap we first establish a comprehensive likelihood theory for the misspecified Generalized Extreme Value (GEV) model under serial dependence. Our results include uniform convergence of the empirical log-likelihood process, contraction rates for the Maximum Likelihood Estimator, and a local asymptotically Gaussian expansion. Building on this foundation, we develop the asymptotic theory of Bayesian inference for the GEV parameters, the extremal index, $T$-time-horizon return levels, and extreme quantiles (Value at Risk). Under general conditions on the prior, we prove posterior consistency, $\sqrt{k}$-contraction rates, Bernstein-von Mises theorems, and asymptotic coverage properties for credible intervals. For inference on the extremal index, we propose an adjusted posterior distribution that corrects for poor coverage exhibited by a naive Bayesian approach. Simulations show excellent inferential performances for the proposed methodology.

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arXiv2025

Predicting hazards of climate extremes: a statistical perspective

Climate extremes such as floods, storms, and heatwaves have caused severe economic and human losses across Europe in recent decades. To support the European Union's climate resilience efforts, we propose a statistical framework for short-to-medium-term prediction of tail risks related to extreme economic losses and fatalities. Our approach builds on Extreme Value Theory and employs the predictive distribution of future tail events to quantify both estimation and aleatoric uncertainty. Using data on EU-wide losses and fatalities from 1980 to 2023, we model extreme events through Peaks Over Threshold methodology and fit Generalised Pareto (GP) and discrete-GP models using an empirical Bayes procedure. Our predictive approach enables a 'What-if' analysis to evaluate hypothetical scenarios beyond observed levels, including potential worst-case outcomes for a precautionary risk assessment of future extreme episodes. To account for a time-varying behavior of extreme losses and fatalities we extend our predictive method using a proportional tail model that allows to handle heteroscedastic extremes over time. Results of our analysis under stationarity and non-stationary settings raise concerns, reinforcing the urgency of integrating predictive tail risk assessment into EU adaptation strategies.

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arXiv2025

Statistical Prediction of Peaks Over a Threshold

In many applied fields, the prediction of more severe events than those already recorded is crucial for safeguarding against potential future calamities. What-if analyses, which evaluate hypothetical scenarios up to the worst-case event, play a key role in assessing the potential impacts of extreme events and guiding the development of effective safety policies. This problem can be analyzed using extreme value theory. We employ the well-established peaks-over-threshold method and describe a comprehensive toolkit to address forecasting needs. We examine an \lq\lq out-of-sample" variable and focus on its conditional probability of exceeding a high threshold, representing the predictive distribution of future extreme peaks. We demonstrate that the generalized Pareto approximation of the corresponding predictive density can be remarkably accurate. We then introduce frequentist methods and a Bayesian approach for estimating this predictive density, enabling the derivation of informative predictive intervals. By leveraging threshold stability, we illustrate how predictions can be reliably extended deep into the tail of the unknown data distribution. We establish the asymptotic accuracy of the proposed estimators and, more importantly, prove that the resulting predictive inference is asymptotically valid. Forecasters satisfying the tail-equivalence property allow to recover widely used risk measures for risk assessment through point forecasts. This insight lays the groundwork for a new perspective that integrates risk assessment into the statistical predictive toolbox. Finally, we extend the prediction framework to the case of linear time series. We apply the proposed predictive tools to two real-world datasets: summer peak temperatures recorded in Milan, Italy, over the past 30 years, and daily negative log-returns of the Dow Jones Industrial Average observed over 30 years.

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arXiv2024

Modeling extremal dependence in multivariate and spatial problems: a practical perspective

From environmental sciences to finance, there is a growing demand for methods that can assess the risks of extreme events beyond those observed in available data. Extrapolating extreme events beyond the range of the data is not obvious. Risk assessments are often further complicated by the need to account for multiple variables simultaneously. Extreme value theory provides important tools for the analysis of multivariate or spatial extreme events, but these are not easily accessible to professionals without appropriate expertise. This article provides a minimal background on multivariate and spatial extremes and gives simple yet thorough instructions on how to analyse them using the R package ExtremalDep. After briefly introducing the statistical methodologies, we focus on road testing the package's toolbox through several real-world applications.

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arXiv2023

Asymptotic theory for Bayesian inference and prediction: from the ordinary to a conditional Peaks-Over-Threshold method

The Peaks Over Threshold (POT) method is the most popular statistical method for the analysis of univariate extremes. Even though there is a rich applied literature on Bayesian inference for the POT, the asymptotic theory for such proposals is missing. Even more importantly, the ambitious and challenging problem of predicting future extreme events according to a proper predictive statistical approach has received no attention to date. In this paper we fill this gap by developing the asymptotic theory of posterior distributions (consistency, contraction rates, asymptotic normality and asymptotic coverage of credible intervals) and prediction within the Bayesian framework in the POT context. We extend this asymptotic theory to account for cases where the focus is on the tail properties of the conditional distribution of a response variable given a vector of random covariates. To enable accurate predictions of extreme events more severe than those previously observed, we derive the posterior predictive distribution as an estimator of the conditional distribution of an out-of-sample random variable, given that it exceeds a sufficiently high threshold. We establish Wasserstein consistency of the posterior predictive distribution under both the unconditional and covariate-conditional approaches and derive its contraction rates. Simulations show the good performances of the proposed Bayesian inferential methods. The analysis of the change in the frequency of financial crises over time shows the utility of our methodology.

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arXiv2023

Strong Convergence of Peaks Over a Threshold

Extreme Value Theory plays an important role to provide approximation results for the extremes of a sequence of independent random variables when their distribution is unknown. An important one is given by the {generalised Pareto distribution} $H_γ(x)$ as an approximation of the distribution $F_t(s(t)x)$ of the excesses over a threshold $t$, where $s(t)$ is a suitable norming function. In this paper we study the rate of convergence of $F_t(s(t)\cdot)$ to $H_γ$ in variational and Hellinger distances and translate it into that regarding the Kullback-Leibler divergence between the respective densities.

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arXiv2023

Marginal expected shortfall inference under multivariate regular variation

Marginal expected shortfall is unquestionably one of the most popular systemic risk measures. Studying its extreme behaviour is particularly relevant for risk protection against severe global financial market downturns. In this context, results of statistical inference rely on the bivariate extreme values approach, disregarding the extremal dependence among a large number of financial institutions that make up the market. In order to take it into account we propose an inferential procedure based on the multivariate regular variation theory. We derive an approximating formula for the extreme marginal expected shortfall and obtain from it an estimator and its bias-corrected version. Then, we show their asymptotic normality, which allows in turn the confidence intervals derivation. Simulations show that the new estimators greatly improve upon the performance of existing ones and confidence intervals are very accurate. An application to financial returns shows the utility of the proposed inferential procedure. Statistical results are extended to a general $β$-mixing context that allows to work with popular time series models with heavy-tailed innovations.

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arXiv2023

Extreme expectile estimation for short-tailed data, with an application to market risk assessment

The use of expectiles in risk management has recently gathered remarkable momentum due to their excellent axiomatic and probabilistic properties. In particular, the class of elicitable law-invariant coherent risk measures only consists of expectiles. While the theory of expectile estimation at central levels is substantial, tail estimation at extreme levels has so far only been considered when the tail of the underlying distribution is heavy. This article is the first work to handle the short-tailed setting where the loss (e.g. negative log-returns) distribution of interest is bounded to the right and the corresponding extreme value index is negative. We derive an asymptotic expansion of tail expectiles in this challenging context under a general second-order extreme value condition, which allows to come up with two semiparametric estimators of extreme expectiles, and with their asymptotic properties in a general model of strictly stationary but weakly dependent observations. A simulation study and a real data analysis from a forecasting perspective are performed to verify and compare the proposed competing estimation procedures.

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arXiv2021

Optimal pooling and distributed inference for the tail index and extreme quantiles

This paper investigates pooling strategies for tail index and extreme quantile estimation from heavy-tailed data. To fully exploit the information contained in several samples, we present general weighted pooled Hill estimators of the tail index and weighted pooled Weissman estimators of extreme quantiles calculated through a nonstandard geometric averaging scheme. We develop their large-sample asymptotic theory across a fixed number of samples, covering the general framework of heterogeneous sample sizes with different and asymptotically dependent distributions. Our results include optimal choices of pooling weights based on asymptotic variance and MSE minimization. In the important application of distributed inference, we prove that the variance-optimal distributed estimators are asymptotically equivalent to the benchmark Hill and Weissman estimators based on the unfeasible combination of subsamples, while the AMSE-optimal distributed estimators enjoy a smaller AMSE than the benchmarks in the case of large bias. We consider additional scenarios where the number of subsamples grows with the total sample size and effective subsample sizes can be low. We extend our methodology to handle serial dependence and the presence of covariates. Simulations confirm that our pooled estimators perform virtually as well as the benchmark estimators. Two applications to real weather and insurance data are showcased.

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arXiv2021

Tail risk inference via expectiles in heavy-tailed time series

Expectiles define the only law-invariant, coherent and elicitable risk measure apart from the expectation. The popularity of expectile-based risk measures is steadily growing and their properties have been studied for independent data, but further results are needed to use extreme expectiles with dependent time series such as financial data. In this paper we establish a basis for inference on extreme expectiles and expectile-based marginal expected shortfall in a general $β$-mixing context that encompasses ARMA, ARCH and GARCH models with heavy-tailed innovations. Simulations and applications to financial returns show that the new estimators and confidence intervals greatly improve on existing ones when the data are dependent.

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arXiv2020

Estimation and uncertainty quantification for extreme quantile regions

Estimation of extreme quantile regions, spaces in which future extreme events can occur with a given low probability, even beyond the range of the observed data, is an important task in the analysis of extremes. Existing methods to estimate such regions are available, but do not provide any measures of estimation uncertainty. We develop univariate and bivariate schemes for estimating extreme quantile regions under the Bayesian paradigm that outperforms existing approaches and provides natural measures of quantile region estimate uncertainty. We examine the method's performance in controlled simulation studies. We illustrate the applicability of the proposed method by analysing high bivariate quantiles for pairs of pollutants, conditionally on different temperature gradations, recorded in Milan, Italy.

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arXiv2020

Consistency of Bayesian Inference for Multivariate Max-Stable Distributions

Predicting extreme events is important in many applications in risk analysis. The extreme-value theory suggests modelling extremes by max-stable distributions. The Bayesian approach provides a natural framework for statistical prediction. Although various Bayesian inferential procedures have been proposed in the literature of univariate extremes and some for multivariate extremes, the study of their asymptotic properties has been left largely untouched. In this paper we focus on a semiparatric Bayesian method for estimating max-stable distributions in arbitrary dimension. We establish consistency of the pertaining posterior distributions for fairly general, well-specified max-stable models, whose margins can be short-, light- or heavy-tailed. We then extend our consistency results to the case where the data come from a distribution lying in a neighbourhood of a max-stable one, which represents the most realistic inferential setting.

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arXiv2020

Joint inference on extreme expectiles for multivariate heavy-tailed distributions

The notion of expectiles, originally introduced in the context of testing for homoscedasticity and conditional symmetry of the error distribution in linear regression, induces a law-invariant, coherent and elicitable risk measure that has received a significant amount of attention in actuarial and financial risk management contexts. A number of recent papers have focused on the behaviour and estimation of extreme expectile-based risk measures and their potential for risk management. Joint inference of several extreme expectiles has however been left untouched; in fact, even the inference of a marginal extreme expectile turns out to be a difficult problem in finite samples. We investigate the simultaneous estimation of several extreme marginal expectiles of a random vector with heavy-tailed marginal distributions. This is done in a general extremal dependence model where the emphasis is on pairwise dependence between the margins. We use our results to derive accurate confidence regions for extreme expectiles, as well as a test for the equality of several extreme expectiles. Our methods are showcased in a finite-sample simulation study and on real financial data.

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arXiv2020

Multivariate Extremes Over a Random Number of Observations

The classical multivariate extreme-value theory concerns the modeling of extremes in a multivariate random sample, suggesting the use of max-stable distributions. In this work, the classical theory is extended to the case where aggregated data, such as maxima of a random number of observations, are considered. We derive a limit theorem concerning the attractors for the distributions of the aggregated data, which boil down to a new family of max-stable distributions. We also connect the extremal dependence structure of classical max-stable distributions and that of our new family of max-stable distributions. By means of an inversion method, we derive a semiparametric composite-estimator for the extremal dependence of the unobservable data, starting from a preliminary estimator of the extremal dependence of the aggregated data. Furthermore, we develop the large-sample theory of the composite-estimator and illustrate its finite-sample performance via a simulation study.

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arXiv2020

Strong Convergence of Multivariate Maxima

It is well known and readily seen that the maximum of $n$ independent and uniformly on $[0,1]$ distributed random variables, suitably standardised, converges in total variation distance, as $n$ increases, to the standard negative exponential distribution. We extend this result to higher dimensions by considering copulas. We show that the strong convergence result holds for copulas that are in a differential neighbourhood of a multivariate generalized Pareto copula. Sklar's theorem then implies convergence in variational distance of the maximum of $n$ independent and identically distributed random vectors with arbitrary common distribution function and (under conditions on the marginals) of its appropriately normalised version. We illustrate how these convergence results can be exploited to establish the almost-sure consistency of some estimation procedures for max-stable models, using sample maxima.

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arXiv2018

Extremal properties of the univariate extended skew-normal distribution

We consider the extremal properties of the highly flexible univariate extended skew-normal distribution. We derive the well-known Mills' inequalities and Mills' ratio for the extended skew-normal distribution and establish the asymptotic extreme-value distribution for the maximum of samples drawn from this distribution.

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arXiv2018

Extremal properties of the multivariate extended skew-normal distribution

The skew-normal and related families are flexible and asymmetric parametric models suitable for modelling a diverse range of systems. We show that the multivariate maximum of a high-dimensional extended skew-normal random sample has asymptotically independent components and derive the speed of convergence of the joint tail. To describe the possible dependence among the components of the multivariate maximum, we show that under appropriate conditions an approximate multivariate extreme-value distribution that leads to a rich dependence structure can be derived.

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