Asymptotic posterior normality of the generalized extreme value distribution

The univariate generalized extreme value (GEV) distribution is the most commonly used tool for analysing the properties of rare events. The ever greater utilization of Bayesian methods for extreme value analysis warrants detailed theoretical investigation, which has thus far been underdeveloped. Even the most basic asymptotic results are difficult to obtain because the GEV fails to satisfy standard regularity conditions. Here, we prove that the posterior distribution of the GEV parameter vector, given an independent and identically distributed sequence of observations, converges to a normal distribution centred at the true parameter. The proof necessitates analysing integrals of the GEV likelihood function over the entire parameter space, which requires considerable care because the support of the GEV density depends on the parameters in complicated ways

Block composite likelihood models for analysis of large spatial datasets

Abstract Large spatial datasets become more common as a result of automatic sensors, remote sensing and the increase in data storage capacity. But large spatial datasets are hard to analyse. Even in the simplest Gaussian situation, parameter estimation and prediction are troublesome because one requires matrix factorization of a large covariance matrix. We consider a composite likelihood construction built on the joint densities of subsets of variables. This composite model thus splits a datasets in many smaller datasets, each of which can be evaluated separately. These subsets of data are combined through a summation giving the final composite likelihood. Massive datasets can be handled with this approach. In particular, we consider a block composite likelihood model, constructed over pairs of spatial blocks. The blocks can be disjoint, overlapping or at various resolution. The main idea is that the spatial blocking should capture the important correlation effects in the data. Estimates for unknown parameters as well as optimal spatial predictions under the block composite model are obtained. Asymptotic variances for both parameter estimates and predictions are computed using Godambe sandwich matrices. The procedure is demonstrated on 2D and 3D datasets with regular and irregular sampling of data. For smaller data size we compare with optimal predictors, for larger data size we discuss and compare various blocking schemes

Semiparametric Estimation of the Shape of the Limiting Multivariate Point Cloud

We propose a model to flexibly estimate joint tail properties by exploiting the convergence of an appropriately scaled point cloud onto a compact limit set. Characteristics of the shape of the limit set correspond to key tail dependence properties. We directly model the shape of the limit set using B\'ezier splines, which allow flexible and parsimonious specification of shapes in two dimensions. We then fit the B\'ezier splines to data in pseudo-polar coordinates using Markov chain Monte Carlo, utilizing a limiting approximation to the conditional likelihood of the radii given angles. By imposing appropriate constraints on the parameters of the B\'ezier splines, we guarantee that each posterior sample is a valid limit set boundary, allowing direct posterior analysis of any quantity derived from the shape of the curve. Furthermore, we obtain interpretable inference on the asymptotic dependence class by using mixture priors with point masses on the corner of the unit box. Finally, we apply our model to bivariate datasets of extremes of variables related to fire risk and air pollution