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Inferences from alarming events.

An extreme event, such as a nuclear accident, an earthquake, a cluster of adverse reactions to a particular drug, or excessive breakdowns of some class of equipment, frequently focuses attention for the first time on an important issue. By then, however, data on the incidence and magnitudes of relevant past events may be unavailable or too costly to reconstruct. Using a simple probability model, we derive methods for drawing statistical inferences based only on the magnitude of the first event noticed and the amount of exposure before this event occurred. We assume that an event is noticed only when its magnitude exceeds some threshold, and we develop methods of inference that are valid even when this threshold is unknown. One tempting but incorrect approach is to treat the magnitude of the observed event as if it were the threshold, forgetting that smaller magnitudes might have been noticed as well. The biases that arise when this mistake is made turn out to be substantial; risks can easily be overstated by a factor of 3.