arXiv subjects

Stefano Rizzelli

Publications and source records attributed to Stefano Rizzelli.

12 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

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

Empirical Bayes in Bayesian learning: understanding a common practice

In applications of Bayesian procedures, once a class of priors has been chosen, it may be tempting to fix the prior's hyperparameters from the data, in an empirical Bayes (EB) fashion, usually by their maximum marginal likelihood estimates (MMLE). This is a quite common but questionable practice, lacking a rigorous theoretical basis. We provide a theoretical framework where this form of EB is regarded as a computational strategy for approximating a genuine Bayesian posterior distribution and prove its general properties for parametric models. While computing the MMLE may still be demanding, we prove novel results that allow us to provide a simple proxy. These results establish the limit behavior of the MMLE in quite general settings, including both identifiable and non-identifiable models - specifically, overfitted mixture models - significantly filling a gap in the literature. Moreover, we study higher order merging, showing that, when not degenerate, the EB posterior approximates at a faster rate an oracle-Bayes posterior distribution based on the prior law that, within the given class of priors, expresses the most information on the true model's parameters. This is a faster approximation than classic Bernstein-von Mises results. Our work provides formal content to common beliefs on this popular practice.

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arXiv2023

Statistical Prediction of Peaks Over a Threshold

In many applied fields it is desired to make predictions with the aim of assessing the plausibility of more severe events than those already recorded to safeguard against calamities that have not yet occurred. This problem can be analysed using extreme value theory. We consider the popular peaks over a threshold method and show that the generalised Pareto approximation of the true predictive densities of both a future unobservable excess or peak random variable can be very accurate. We propose both a frequentist and a Bayesian approach for the estimation of such predictive densities. We show the asymptotic accuracy of the corresponding estimators and, more importantly, prove that the resulting predictive inference is asymptotically reliable. We show the utility of the proposed predictive tools analysing extreme temperatures in Milan in Italy.

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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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arXiv2022

Empirical Bayes inference for the block maxima method

The block maxima method is one of the most popular approaches for extreme value analysis with independent and identically distributed observations in the domain of attraction of an extreme value distribution. The lack of a rigorous study on the Bayesian inference in this context has limited its use for statistical analysis of extremes. In this paper we propose an empirical Bayes procedure for inference on the block maxima law and its related quantities. We show that the posterior distributions of the tail index of the data distribution and of the return levels (representative of future extreme episodes) satisfy a number of important theoretical properties. These guarantee the reliability of posterior-based inference and extend to the posterior predictive distribution, the key tool in Bayesian probabilistic forecasting. Posterior computations are readily obtained via an efficient adaptive Metropolis-Hasting type of algorithm. Simulations show its excellent inferential performances already with modest sample sizes. The utility of our proposal is showcased analysing extreme winds generated by hurricanes in the Atlantic basin.

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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

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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