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The complexity of forecast testing. (English) Zbl 1160.91396
Summary: Consider a weather forecaster predicting a probability of rain for the next day. We consider tests that, given a finite sequence of forecast predictions and outcomes, will either pass or fail the forecaster. Sandroni showed that any test which passes a forecaster who knows the distribution of nature can also be probabilistically passed by a forecaster with no knowledge of future events. We look at the computational complexity of such forecasters and exhibit a linear-time test and distribution of nature such that any forecaster without knowledge of the future who can fool the test must be able to solve computationally difficult problems. Thus, unlike Sandroni’s work, a computationally efficient forecaster cannot always fool this test independently of nature.

MSC:
91B82 Statistical methods; economic indices and measures
62M09 Non-Markovian processes: estimation
60G25 Prediction theory (aspects of stochastic processes)
91A26 Rationality and learning in game theory
62G10 Nonparametric hypothesis testing
62P30 Applications of statistics in engineering and industry; control charts
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