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Extreme value theory sits alone in the statistical sciences. It is the branch of statistics devoted to the inference of extreme events in random processes, and differs from most areas of statistics in modelling rare, rather than typical, behaviour. A common aim is to estimate what future extreme levels of a process might be expected based on a historical series of observations. As such, the methodology is widely used in engineering applications that need an assessment of extreme environmental conditions: for example, sea levels, wind speeds or river flow (Coles 2001, chapter 1). Another recent area of application is to financial markets, for which calculations on the plausibility of large returns (postitive or negative) are helpful, and even a statutory requirement for many banks (e.g. Embrechts et al. 1998).

Volcanic events are among the most explosive on Earth. A recent study (Mason et al. 2004) has compared the potential damage from large volcanoes with that of other potentially catastrophic events, such as a meteor strike. At least in an informal sense therefore, they are extreme events, and it seems reasonable to hope that extreme value theory can make a useful contribution to their analysis. In this paper we explore this possibility. We take the recent history of large eruptive volcanoes, and apply extreme value techniques to obtain estimates of the probability of future extreme eruption events at different levels of magnitude. Such inferences are of general volcanologi-cal interest, but they are also directly .

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