A unified conditional frequentist and Bayesian test for fixed and sequential simple hypothesis testing. (English) Zbl 0824.62002

Berger, James O.; Brown, Lawrence D.; Wolpert, Robert L.

Ann. Stat. 22, No. 4, 1787-1807 (1994).

Summary: Preexperimental frequentist error probabilities are arguably inadequate, as summaries of evidence from data, in many hypothesis-testing settings. The conditional frequentist may respond to this by identifying certain subsets of the outcome space and reporting a conditional error probability, given the subset of the outcome space in which the observed data lie. Statistical methods consistent with the likelihood principle, including Bayesian methods, avoid the problem by a more extreme form of conditioning.
We prove that the conditional frequentist method can be made exactly equivalent to the Bayesian’s in simple versus simple hypothesis testing: specifically, we find a conditioning strategy for which the conditional frequentist’s reported conditional error probabilities are the same as the Bayesian’s posterior probabilities of error. A conditional frequentist who uses such a strategy can exploit other features of the Bayesian approach – for example, the validity of sequential hypothesis tests (including versions of the sequential probability ratio test, or SPRT) even if the stopping rule is incompletely specified.

MSC:

62A01	Foundations and philosophical topics in statistics
62L10	Sequential statistical analysis

Keywords:

likelihood principle; Bayes factor; likelihood ratio; significance; Type I error; stopping rule principle; conditional frequentist method; conditioning strategy; conditional error probabilities; posterior probabilities of error; sequential probability ratio test