Jonathan A. Tawn

Dependence Measures for Extreme Value Analyses

Quantifying dependence is a central theme in probabilistic and statistical methods for multivariate extreme values. Two situations are possible: one where, in a limiting sense, the extremes are dependent; the other where, in the same sense, the extremes are independent. This paper comprises an overview of the principal issues through a unified approach which encompasses both these situations. Novel diagnostic measures for dependence are also developed which provide complementary information about different aspects of extremal dependence. The paper is written in an elementary style, with the methodology illustrated by application to theoretical examples and typical data-sets. These data-sets and the S-plus functions used for the analyses are available online.

Modelling Dependence within Joint Tail Regions

Standard approaches for modelling dependence within joint tail regions are based on extreme value methods which assume max-stability, a particular form of joint tail dependence. We develop joint tail models based on a broader class of dependence structure which provides a natural link between max-stable models and weaker forms of dependence including independence and negative association. This approach overcomes many of the problems that are encountered with standard methods and is the basis for a Poisson process representation that generalizes existing bivariate results. We apply the new techniques to simulated and environmental data, and demonstrate the marked advantage that the new approach offers for joint tail extrapolation.

A Bayesian analysis of extreme rainfall data

SUMMARY Understanding and quantifying the behaviour of a rainfall process at extreme levels has important applications for design in civil engineering. As in the extremal analysis of any environmental process, estimates often are required of the probability of events that are rarer than those already recorded. As data on extremes are scarce, all available sources of information should be used in inference. Consequently, research has focused on the development of techniques that make optimal use of available data. I n this paper a daily rainfall series is analysed within a Bayesian framework, illustrating how the careful elicitation of prior expert information can supplement data and lead to improved estimates of extremal behaviour. For example, using the prior knowledge of an expert hydrologist, a Bayesian 95% interval estimate of the 1 00-year return level for daily rainfall is found to be approximately half of the width of the corresponding likelihoodbased confidence interval,

Statistical methods for multivariate extremes - an application to structural design

For many structural design problems univariate extreme value theory is applied to quantify the risk of failure due to extreme levels of some environmental process. In practice, many forms of structure fail owing to a combination of various processes at extreme levels. Recent developments in statistical methodology for multivariate extremes enable the modelling of such behaviour. The aim of this paper is to demonstrate how these ideas can be exploited as part of the design process

Flood frequency estimation by continuous simulation for a gauged upland catchment (with uncertainty)

This paper explores the possibility of deriving frequency distributions of extreme discharges by continuous simulation. The rainfall-runoff model TOPMODEL is applied within the Generalised Likelihood Uncertainty Estimation (GLUE) framework to the River Wye catchment, Plynlimon, Wales, using a 21-year period of rainfall and discharge observations. Rejection of non-behavioural parameter sets is achieved through an evaluation of both annual maximum discharge and continuous hydrograph simulation. Annual maximum peak timings and rankings are also considered. It is demonstrated that, within the prescribed limits, TOPMODEL can adequately achieve both flood frequency and continuous simulation modelling goals. Extension of the flood frequency estimations beyond the upper limit of the observed series is attained through the coupling of behavioural TOPMODEL sets with those of a stochastic rainfall generator for 1000-year simulation periods using hourly time steps. The rainfall model is conditioned on the observed rainfall frequency statistics for different storm duration classes, also within the GLUE framework.

An extreme-value theory model for dependent observations

Abstract Modelling extreme values from an environmental time series requires an extreme-value theory model which can handle dependent observations. A method of filtering the original time series to obtain independent extremes is presented. The resulting extremes are then modelled using an extension of suggested ideas∗. Here the limiting joint Generalized Extreme Value distribution for the r largest order statistics is considered; whereas others ∗∗ used the corresponding Gumbel distribution. Various tests of fit of the model are discussed, with a detailed analysis of how to test for dependence between extremes in the original sequence. An additional method of using data from neighbouring sites to improve the estimation is suggested. The procedures and tests are illustrated by an application to the sea levels at Lowestoft and Great Yarmouth.

The joint probability of waves and water levels in coastal engineering design

On coasts with high tidal ranges, or subject to high surges, both still water levels and waves can be important in assessing flood risk; their relative importance depends on location and on the type of sea defence. The simultaneous occurrence of large waves and a high still water level is therefore important in estimating their combined effect on sea defences. Wave period can also be important in assessing run-up and overtopping, and so it is useful also to have information on the joint distribution of wave height and period. Unless the variables are either completely independent or completely dependent, multivariate extremes are difficult to predict directly from observational data, as there may be too few events of the relevant type amongst the observations. In the past, the fitting and extrapolation of the dependence functions between the variables has often involved complicated and/or subjective approaches. This paper presents a method for joint probability analysis, using a Monte Carlo simulation approach, based on distributions fitted to water level, wave height and wave steepness, and to the dependence between them.

Diagnostics for dependence within time series extremes

Summary. The analysis of extreme values within a stationary time series entails various assumptions concerning its long- and short-range dependence. We present a range of new diagnostic tools for assessing whether these assumptions are appropriate and for identifying structure within extreme events. These tools are based on tail characteristics of joint survivor functions but can be implemented by using existing estimation methods for extremes of univariate independent and identically distributed variables. Our diagnostic aids are illustrated through theoretical examples, simulation studies and by application to rainfall and exchange rate data. On the basis of these diagnostics we can explain characteristics that are found in the observed extreme events of these series and also gain insight into the properties of events that are more extreme than those observed.

Estimating probabilities of extreme sea-levels

A key problem in the design of sea defences is the estimation of quantiles of the distribution of annual maximum hourly sea‐levels. Traditional statistical analyses fail to exploit the considerable knowledge of the astronomical tidal component of the sea; consequently the corresponding results are highly site specific. Using results from extreme value theory an ad hoc method developed by oceanographers to overcome this problem is revised. The method is illustrated with data from three sites on the east coast of England which exhibit widely differing characteristics.

Statistics of Multivariate Extremes

Summary The paper is concerned with statistical aspects of multivariate extreme value distributions. The family is infinite dimensional, so direct parametric estimation is not possible. We describe both nonparametric and parametric approaches, the latter being based on parametric subfamilies. Some problems connected with maximum likelihood estimators are discussed, and solutions proposed. In the nonparametric cases, the main method is an adaptation of the kernel method for density estimation. The detailed discussion is restricted to the bivariate case, but we also outline how the methods might be extended to higher dimensions.