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.     

Extended generalised Pareto models for tail estimation

The most popular approach in extreme value statistics is the modelling of threshold exceedances using the asymptotically motivated generalised Pareto distribution. This approach involves the selection of a high threshold above which the model fits the data well. Sometimes, few observations of a measurement process might be recorded in applications and so selecting a high quantile of the sample as the threshold leads to almost no exceedances. In this paper we propose extensions of the generalised Pareto distribution that incorporate an additional shape parameter while keeping the tail behaviour unaffected. The inclusion of this parameter offers additional structure for the main body of the distribution, improves the stability of the modified scale, tail index and return level estimates to threshold choice and allows a lower threshold to be selected. We illustrate the benefits of the proposed models with a simulation study and two case studies.     

Weight-preserving simulated tempering

Statistics and Computing - Simulated tempering is a popular method of allowing MCMC algorithms to move between modes of a multimodal target density $$\pi $$. One problem with simulated tempering...     

Ordered multivariate extremes

Multivariate extreme value models and associated statistical methods are developed for vector observations whose components are subject to an order restriction. The approach extends the multivariate threshold methodology of Coles and Tawn, Joe and co-workers and Smith and co-workers. The results are illustrated by an analysis of extreme rainfalls of different durations, and by a study of the problem of linking a long series of daily rainfall extremes with a partially overlapping shorter series of hourly extremes.     

Extreme values in the dock

On 9th September 1980 the bulk carrier M. V. Derbyshire sank in the Pacific Ocean, around 350 miles south-east of Japan, when she was caught in Typhoon Orchid while transporting iron ore from Canada to Japan. All 44 people on board died. Janet Heffernan and Jonathan Tawn describe their involvement as statistical expert witnesses in the Investigation of the sinking and their experience of giving evidence in court, outlining their statistical analysis which provided new insight into the circumstances surrounding the loss of the ship.     

Modification of Pickands' Dependence Function for Ordered Bivariate Extreme Distribution

We study the characteristics of the Pickands' dependence function for bivariate extreme distribution for minima, BEVM, when considering the stochastics ordering of the two variables, X < Y. The existing Pickand's dependence function terminologies and theories are modified to suit the dependence functions of extreme minimum cases. The main result is the introduction of the restricted logistic dependence function, A _RL , and the restricted exponential function, V _RL (x, y).     

A semi-parametric model for multivariate extreme values

Threshold methods for multivariate extreme values are based on the use of asymptotically justified approximations of both the marginal distributions and the dependence structure in the joint tail. Models derived from these approximations are fitted to a region of the observed joint tail which is determined by suitably chosen high thresholds. A drawback of the existing methods is the necessity for the same thresholds to be taken for the convergence of both marginal and dependence aspects, which can result in inefficient estimation. In this paper an extension of the existing models, which removes this constraint, is proposed. The resulting model is semi-parametric and requires computationally intensive techniques for likelihood evaluation. The methods are illustrated using a coastal engineering application.     

Statistics for Extreme Sea Currents

Estimates of various characteristics of extreme sea currents, such as speeds and their directions, are required when designing offshore structures. This paper extends standard statistical methods for extreme values to handle the directionality, temporal dependence and tidal non-stationarity that are present in sea current extremes. The methods are applied to a short period of data from the Inner Dowsing Light Tower in the North Sea. Substantial benefits, over existing methods, are obtained from our analysis of the sea current by decomposing it into tide and surge currents. In particular, we find that at the Inner Dowsing the strong directionality in extreme sea current speeds is completely explained by the tidal current and directionality in the non-extreme surge currents. This finding aids model fitting and extrapolation.