Statistical Modeling of Extreme Events

Abstract

Extreme value (EV) analysis involves the estimation of the probability of events that are unusually large or small. EV methods have a wide range of application from modeling extreme wave heights and water levels in hydrology, structural engineering to share price return levels in finances. In case of univariate independent and identically distributed (i.i.d.) random variables, a number of statistical models do exist in literature. However, for dependent and non-stationary multivariate extremes the development of different statistical model remains an ongoing area of research. Most of my reading and work has been motivated by the application of the EV analysis in designing oil and gas producing facilities, off-shore or on-shore for extreme ocean environments. It becomes essential to model covariate effects (wave directions, seasons etc.) for the data observed over the years in the oceans. We begin with the study of different existing models which incorporate these covariate effects such as conditional extremes model (Heffernan and Tawn, 2004)and Non-stationary conditional extremes (NSCE) model (Raghupathi et al. 2016). However, the application of the frequentist NSCE model seems to be computationally challenging and expensive. So, next we study a piece-wise model for a sample of peaks over threshold which is non-stationary with respect to multidimensional co-variates, estimated using a computationally efficient Bayesian inference. We then study the convergence diagnostics for the Markov Chain Monte Carlo (MCMC) procedure used in estimation of the desired Bayesian model by application on synthetic data. Most importantly, we study the problem of threshold estimation using the Bayesian inference in application for one covariate (direction).