Line mesh distributions: An alternative approach for multivariate environmental extremes

Abstract

Copulas and other multivariate models can give joint exceedance probabilities for multivariate events in the naturalenvironment. However, the choice of the most appropriate multivariate model may not always be evident in the absence ofknowledge of dependence structures. A simple nonparametric alternative is to approximate multivariate dependenciesusing ‘‘line mesh distributions’’, introduced here as a data-based finite mixture of univariate distributions defined on a meshof L =C(m, 2) lines extending through Euclidean n-space. That is, m data points in n-space define a total of L lines, whereC() denotes the binomial coefficient. The utilitarian simplicity of the method has attraction for joint exceedance proba-bilities because just the data and a single bandwidth parameter within the 0, 1 interval are needed to define a line meshdistribution. All bivariate planes in these distributions have the same Pearson correlation coefficients as the correspondingdata. Marginal means and variances are similarly preserved. Using an example from the literature, a 5-parameter bivariateGumbel model is replaced with a 1-parameter line mesh distribution. A second illustration for three dimensions applies linemesh distributions to data simulated from a trivariate copula.