Threshold modelling of spatially dependent non-stationary extremes with application to hurricane-induced wave heights Rejoinder

In environmental applications it is common for the extremes of a variable to be non-stationary, varying systematically in space, time or with the values of covariates. Multi-site datasets are common, and in such cases there is likely to be non-negligible inter-site dependence. We consider applications in which multi-site data are used to infer the marginal behaviour of the extremes at individual sites, while adjusting for inter-site dependence. For reasons of statistical efficiency, it is standard to model exceedances of a high threshold. Choosing an appropriate threshold can be problematic, particularly if the extremes are non-stationary. We propose a method for setting a covariate-dependent threshold using quantile regression. We consider how the quantile regression model and extreme value models fitted to threshold exceedances should be parameterized, in order that they are compatible. We adjust estimates of uncertainty for spatial dependence using methodology proposed recently. These methods are illustrated using time series of storm peak significant wave heights from 72 sites in the Gulf of Mexico. A simulation study illustrates the applicability of the proposed methodology more generally. Copyright (C) 2011 John Wiley & Sons, Ltd.