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

Clément Dombry, Simone A. Padoan, Stefano Rizzelli

TL;DR

This work develops a rigorous Bayesian asymptotic theory for Peaks Over Threshold (POT) methods in univariate extremes and extends it to conditional extremes via a proportional tail model. It establishes posterior consistency, $\sqrt{k}$-contraction rates, a Bernstein–von Mises limit, and Wasserstein-consistent posterior predictive distributions, addressing the intrinsic GP misspecification and threshold-dependence challenges. For extremes conditional on covariates, it introduces a Dirichlet Process prior on the covariate distribution to model the scedasis function $c_0(x)$, proving contraction and BvM results and enabling predictive inference for conditional extreme quantiles through the posterior predictive distribution. Simulation studies and a financial-crisis application demonstrate good finite-sample performance, with accurate quantile forecasts and the ability to test covariate effects on extremes. Overall, the paper provides a principled, predictive Bayesian framework for both unconditional and covariate-adjusted extreme events, with practical implications for risk assessment and forecasting of future extreme occurrences.

Abstract

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.

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

TL;DR

This work develops a rigorous Bayesian asymptotic theory for Peaks Over Threshold (POT) methods in univariate extremes and extends it to conditional extremes via a proportional tail model. It establishes posterior consistency, -contraction rates, a Bernstein–von Mises limit, and Wasserstein-consistent posterior predictive distributions, addressing the intrinsic GP misspecification and threshold-dependence challenges. For extremes conditional on covariates, it introduces a Dirichlet Process prior on the covariate distribution to model the scedasis function , proving contraction and BvM results and enabling predictive inference for conditional extreme quantiles through the posterior predictive distribution. Simulation studies and a financial-crisis application demonstrate good finite-sample performance, with accurate quantile forecasts and the ability to test covariate effects on extremes. Overall, the paper provides a principled, predictive Bayesian framework for both unconditional and covariate-adjusted extreme events, with practical implications for risk assessment and forecasting of future extreme occurrences.

Abstract

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.
Paper Structure (18 sections, 14 theorems, 49 equations, 3 figures, 1 table)

This paper contains 18 sections, 14 theorems, 49 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Statistical model and its challenges
  3. Objectives and contributions
  4. Workflow
  5. The Peaks-Over-Threshold approach
  6. Background
  7. Empirical log-likelihood process asymptotics
  8. Asymptotic theory of the posterior distribution
  9. Asymptotic theory of predictive distribution
  10. Extreme regression
  11. Proportional tail model
  12. Asymptotic theory of the posterior distribution
  13. Asymptotic theory of predictive distribution
  14. Simulation experiments
  15. Posterior distribution computation
  16. ...and 3 more sections

Key Result

Proposition 2.1

Under Condition cond:second_order there exist $\varepsilon_0>0$ and constants $c_1,c_2,c_3>0$ such that the following three properties hold with probability tending to $1$ as $n\to\infty$:

Figures (3)

  • Figure 1: Estimated power functions. Lines report the empirical proportion of simulated samples under $H_1: P_0^*\neq P_0$ that rejected $H_0:P_0^* = P_0$ as a function $\beta$. Dotted red horizontal line is the $5\%$ significance level of the test.
  • Figure 2: Boxplot of $\overline{c}_n(x)-c_0(x)$ (top panels) and $\log(\overline{Q}_{x,n})-\log(F^{(0)\leftarrow}_x(0.001))$ (bottom panels), obtained with the scedasis model (i)-(iii) and different covariate's distributions and setting the covariate value $x=0.1,0.5,0.9$.
  • Figure 3: S&P 500 return estimation results. EVI Posterior distribution (left panel), estimated scedasis function (middle panel) and loss-returns with conditional extreme quantile estimates and predictive intervals superimposed.

Theorems & Definitions (20)

  • Proposition 2.1
  • Corollary 2.2
  • Remark 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Proposition 2.6
  • Remark 2.7
  • Example : Informative data dependent prior
  • Example : Non-informative improper prior
  • Theorem 2.8
  • ...and 10 more