Over a decade ago, point rainfall models based upon Poisson cluster processes were developed by Rodriguez-Iturbe, Cox and Isham. Two types of point process models were envisaged: the Bartlett±Lewis and the Neyman±Scott rectangular pulse models. Recent developments are reviewed here, including a number of empirical studies. The parameter estimation problem is addressed for both types of Poisson-cluster based models. The multiplicity of parameters which can be obtained for a given data set using the method of moments is illustrated and two approaches to ®nding a best set of parameters are presented. The use of a proper ®tting method will allow for the problems encountered in regionalisation to be adequately dealt with. Applications of the point process model to¯ood design are discussed and ®nally, results for a model with dependent cell depth and duration are given. Taking into account the spatial features of rainfall, three multi-site models are presented and compared. They are all governed by a master Poisson process of storm origins and have a number of cell origins associated with each storm origin. The three models differ as to the type of dependence structure between the cell characteristics at different sites. Analytical properties are presented for these models and their ability to represent the spatial structure of a set of raingauge data in the South-West of England is examined. Continuous spatial-temporal models are currently being developed and results are presented for a model in which storm centres arrive in a homogeneous Poisson process in

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.

A spatial-temporal model of rainfall is studied in which storms arrive in a Poisson process in time, each storm giving rise to a random number of elliptical rain cells. Each rain cell moves with a random velocity for a random time before terminating. Rain is deposited by the cell at a random intensity which is constant over the area of the cell and over its lifetime. The main properties of this model are studied analytically where possible. Further properties and the aggregation of model properties over space for direct comparison with rainfall radar data require the numerical evaluation of integrals.

A simple statistical model is used to partition uncertainty from different sources, in projections of future climate from multimodel ensembles. Three major sources of uncertainty are considered: the choice of climate model, the choice of emissions scenario, and the internal variability of the modeled climate system. The relative contributions of these sources are quantified for mid-and late-twenty-first-century climate projections, using data from 23 coupled atmosphere-ocean general circulation models obtained from phase 3 of the Coupled Model Intercomparison Project (CMIP3). Similar investigations have been carried out recently by other authors but within a statistical framework for which the unbalanced nature of the data and the small number (three) of scenarios involved are potentially problematic. Here, a Bayesian analysis is used to overcome these difficulties. Global and regional analyses of surface air temperature and precipitation are performed. It is found that the relative contributions to uncertainty depend on the climate variable considered, as well as the region and time horizon. As expected, the uncertainty due to the choice of emissions scenario becomes more important toward the end of the twenty-first century. However, for midcentury temperature, model internal variability makes a large contribution in high-latitude regions. For midcentury precipitation, model internal variability is even more important and this persists in some regions into the late century. Implications for the design of climate model experiments are discussed.

Summary Design conditions for marine structures are typically informed by threshold‐based extreme value analyses of oceanographic variables, in which excesses of a high threshold are modelled by a generalized Pareto distribution. Too low a threshold leads to bias from model misspecification, and raising the threshold increases the variance of estimators: a bias–variance trade‐off. Many existing threshold selection methods do not address this trade‐off directly but rather aim to select the lowest threshold above which the generalized Pareto model is judged to hold approximately. In the paper Bayesian cross‐validation is used to address the trade‐off by comparing thresholds based on predictive ability at extreme levels. Extremal inferences can be sensitive to the choice of a single threshold. We use Bayesian model averaging to combine inferences from many thresholds, thereby reducing sensitivity to the choice of a single threshold. The methodology is applied to significant wave height data sets from the northern North Sea and the Gulf of Mexico.

A crucial aspect of threshold-based extreme value analyses is the level at which the threshold is set. For a suitably high threshold asymptotic theory suggests that threshold excesses may be modelled by a generalized Pareto distribution. A common threshold diagnostic is a plot of estimates of the generalized Pareto shape parameter over a range of thresholds. The aim is to select the lowest threshold above which the estimates are judged to be approximately constant, taking into account sampling variability summarized by pointwise confidence intervals. This approach doesn't test directly the hypothesis that the underlying shape parameter is constant above a given threshold, but requires the user subjectively to combine information from many dependent estimates and confidence intervals. We develop tests of this hypothesis based on a multiple-threshold penultimate model that generalizes a two-threshold model proposed recently. One variant uses only the model fits from the traditional parameter stability plot. This is particularly beneficial when many datasets are analysed and enables assessment of the properties of the test on simulated data. We assess and illustrate these tests on river flow rate data and 72 series of significant wave heights.

Some of the many statistical aspects of floods are discussed and some new results are given on a number of features. The probability theory of extreme values is briefly reviewed and its potential for applications outlined. The peaks-over-threshold method of estimation is compared with the direct use of annual maxima and some theoretical comparisons of efficiency are given. The effect of trend on the distribution of maxima is analysed in terms of a simple theoretical model distinguishing the effects of trend in mean level and of trend in dispersion. An empirical Bayes method for pooling information from a number of sources is described and illustrated, and related to the procedures recommended in the Flood estimation handbook. In the final section, a range of further issues is outlined.

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