  The prediction of extreme events in time series is a fundamental problem
arising in many financial, scientific, engineering, and other applications. We
begin by establishing a general Neyman-Pearson-type characterization of optimal
extreme event predictors in terms of density ratios. This yields new insights
and several closed-form optimal extreme event predictors for additive models.
These results naturally extend to time series, where we study optimal extreme
event prediction for heavy-tailed autoregressive and moving average models.
Using a uniform law of large numbers for ergodic time series, we establish the
asymptotic optimality of an empirical version of the optimal predictor for
autoregressive models. Using multivariate regular variation, we also obtain
expressions for the optimal extremal precision in heavy-tailed infinite moving
averages, which provide theoretical bounds on the ability to predict extremes
in this general class of models. The developed theory and methodology is
applied to the important problem of solar flare prediction based on the
state-of-the-art GOES satellite flux measurements of the Sun. Our results
demonstrate the success and limitations of long-memory autoregressive as well
as long-range dependent heavy-tailed FARIMA models for the prediction of
extreme solar flares.
