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Goodness-of-fit tests based on the empirical Wasserstein distance are proposed for simple and composite null hypotheses involving general multivariate distributions. For group families, the procedure is to be implemented after preliminary reduction of the data via invariance. This property allows for calculation of exact critical values and p-values at finite sample sizes. Applications include testing for location–scale families and testing for families arising from affine transformations, such as elliptical distributions with given standard radial density and unspecified location vector and scatter matrix. A novel test for multivariate normality with unspecified mean vector and covariance matrix arises as a special case. For more general parametric families, we propose a parametric bootstrap procedure to calculate critical values. The lack of asymptotic distribution theory for the empirical Wasserstein distance means that the validity of the parametric bootstrap under the null hypothesis remains a conjecture. Nevertheless, we show that the test is consistent against fixed alternatives. To this end, we prove a uniform law of large numbers for the empirical distribution in Wasserstein distance, where the uniformity is over any class of underlying distributions satisfying a uniform integrability condition but no additional moment assumptions. The calculation of test statistics boils down to solving the well-studied semi-discrete optimal transport problem. Extensive numerical experiments demonstrate the practical feasibility and the excellent performance of the proposed tests for the Wasserstein distance of order p = 1 and p = 2 and for dimensions at least up to d = 5. The simulations also lend support to the conjecture of the asymptotic validity of the parametric bootstrap.

A Markov tree is a probabilistic graphical model for a random vector by which conditional independence relations between variables are encoded via an undirected tree and each node corresponds to a variable. One possible max-stable attractor for such a model is a Hüsler-Reiss extreme value distribution whose variogram matrix inherits its structure from the tree, each edge contributing one free dependence parameter. Even if some of the variables are latent, as can occur on junctions or splits in a river network, the underlying model parameters are still identifiable if and only if every node corresponding to a missing variable has degree at least three. Three estimation procedures, based on the method of moments, maximum composite likelihood, and pairwise extremal coeffcients, are proposed for usage on multivariate peaks over thresholds data. The model and the methods are illustrated on a dataset of high water levels at several locations on the Seine network. The structured Hüsler-Reiss distribution is found to fit the observed extremal dependence well, and the fitted model confirms the importance of ow-connectedness for the strength of dependence between high water levels, even for locations at large distance apart.

Describing the complex dependence structure of extreme phenomena is particularly challenging. To tackle this issue we develop a novel statistical method that describes extremal dependence taking advantage of the inherent tree-based dependence structure of the max-stable nested logistic distribution, and that identifies possible clusters of extreme variables using reversible jump Markov chain Monte Carlo techniques. Parsimonious representations are achieved when clusters of extreme variables are found to be completely independent. Moreover, we significantly decrease the computational complexity of full likelihood inference by deriving a recursive formula for the likelihood function of the nested logistic model. The method's performance is verified through extensive simulation experiments which also compare different likelihood procedures. The new methodology is used to investigate the dependence relationships between extreme concentrations of multiple pollutants in California and how these concentrations are related to extreme weather conditions. Overall, we show that our approach allows for the representation of complex extremal dependence structures and has valid applications in multivariate data analysis, such as air pollution monitoring, where it can guide policymaking

Many applications in risk analysis, especially in environmental sciences, require the estimation of the dependence among multivariate maxima. A way to do this is by inferring the Pickands dependence function of the underlying extreme-value copula. A nonparametric estimator is constructed as the sample equivalent of a multivariate extension of the madogram. Shape constraints on the family of Pickands dependence functions are taken into account by means of a representation in terms of a specific type of Bernstein polynomials. The large-sample theory of the estimator is developed and its finite-sample performance is evaluated with a simulation study. The approach is illustrated by analyzing clusters consisting of seven weather stations that have recorded weekly maxima of hourly rainfall in France from 1993 to 2011.