Bayesian analysis of nonstationary extremes

Abstract

Extreme value theory (EVT) provides theoretical foundation to assess the probability of rare and extreme events. This thesis provides novel Bayesian semiparametric models for capturing the behaviour of extreme events and extremal dependence with an emphasis on covariate-adjusted variation.

We fist develop Bayesian semiparametric inferences for heteroscedastic extremes. The proposed model is based on an extreme value index regression and a proportional tails model, and assesses how the magnitude and frequency of the extreme values evolve according to a covariate. We present inference methods about the extreme value index and the scedasis density function in the global setting, and then extend the methods to the conditional setting using the dependent Bernstein{Dirichlet process. Our methods are applied to extreme currency demand in Portugal. The signatures of the fitted scedasis densities of currency demand across di erent denominations reveal some interesting insights into the dynamics governing currency demand during periods of economic stress.

We extend our scope of analysis into multivariate extremes and propose a Bayesian model that learns about the dynamics governing joint extreme values over time. Dual time-varying measures for pairwise extremal dependence are proposed to assess the strength of dependence of joint extreme values over time under the settings of asymptotic dependence and asymptotic independence. These measures are modelled via a suitable class of generalised additive models (GAMs) and the dynamics of underlying extremal dependence structure is tracked by Bayesian penalised splines. The application to major stock market indices reveals complex patterns of extremal dependence between countries over the last three decades, including transitions from asymptotic dependence to asymptotic independence.

For a full description of dependence structure between multivariate extremes, a Bayesian inference for extreme-value copulae is proposed along with the dependent Bernstein Dirichlet process prior with the mean constraints. The proposed method rst recovers conditional angular densities (angular surface) and then estimate the extreme-value copula which represent the extremal dependence between two random variables conditioning on a covariate. Simulation studies and an application to the extremal dependence between cryptocurrencies over time, are also provided to evaluate our methods. Finally, we discuss future extensions in the direction of a general and exible modelling framework.