BAYESIAN FORECASTING OF EXTREME VALUES IN AN EXCHANGEABLE SEQUENCE

This article develops new theory and methodology for the forecasting o f extreme and/or record values in an exchangeable sequence of random v ariables. The Hill tail index estimator for long-tailed distributions is modified so as to be appropriate for prediction of future variables . Some basic issues regarding the use of finite, versus infinite ideal ized models, are discussed. It is shown that the standard idealized lo ng-tailed model with tail index alpha less-than-or-equal-to 2 can lead to unrealistic predictions if the observable data is assumed to be un bounded. However, if the model is instead viewed as valid only for som e appropriate finite domain, then it is compatible with, and leads to sharper versions of, sensible methods for prediction. In particular, t he prediction of the next record value is then at most a few multiples of the current record. It is argued that there is no more reason to e schew posterior expectations for forecasting in the context of long-ta iled distributions than to do so in any other context, such as in the many applications where expectations are routinely used for scientific inference and decision-making. Computer simulations are used to demon strate the effectiveness of the methodology, and its use in forecastin g is illustrated.