Bayesian Forecasting of Extreme Values in an Exchangeable Sequence

This article develops new theory and methodology for the forecasting of extreme and/or record values in an exchangeable sequence of random variables. 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 idealized models, are discussed. It is shown that the standard idealized long-tailed model with tail index α ≤ 2 can lead to unrealistic predictions if the observable data is assumed to be unbounded. However, if the model is instead viewed as valid only for some appropriate finite domain, then it is compatible with, and leads to sharper versions of, sensible methods for prediction. In particular, the 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 eschew posterior expectations for forecasting in the context of long-tailed 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 demonstrate the effectiveness of the methodology, and its use in forecasting is illustrated.