The Annals of Statistics

Evaluating probability forecasts

Tze Leung Lai, Shulamith T. Gross, and David Bo Shen

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Probability forecasts of events are routinely used in climate predictions, in forecasting default probabilities on bank loans or in estimating the probability of a patient’s positive response to treatment. Scoring rules have long been used to assess the efficacy of the forecast probabilities after observing the occurrence, or nonoccurrence, of the predicted events. We develop herein a statistical theory for scoring rules and propose an alternative approach to the evaluation of probability forecasts. This approach uses loss functions relating the predicted to the actual probabilities of the events and applies martingale theory to exploit the temporal structure between the forecast and the subsequent occurrence or nonoccurrence of the event.

Article information

Ann. Statist. Volume 39, Number 5 (2011), 2356-2382.

First available in Project Euclid: 30 November 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G42: Martingales with discrete parameter 62P99: None of the above, but in this section
Secondary: 62P05: Applications to actuarial sciences and financial mathematics

Forecasting loss functions martingales scoring rules


Lai, Tze Leung; Gross, Shulamith T.; Shen, David Bo. Evaluating probability forecasts. Ann. Statist. 39 (2011), no. 5, 2356--2382. doi:10.1214/11-AOS902.

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