Reference priors when the stopping rule depends on the parameter of interest

Priors related to Jeffreys's prior and the reference priors of Berger and Bernardo are developed for experiments where a stopping rule is given. These noninformative priors depend on a function of the expected stopping time. In a one-parameter experiment with a stopping rule, the Jeffreys prior is shown to be improved by taking the stopping rule into account, based on a criterion using frequentist coverage probabilities. In a multiparameter experiment, however, the Jeffreys prior depends on the stopping rule in an inappropriate fashion. In the latter case the reference prior effectively accounts for the stopping rule. The motivation for this work was in part the statistical importance of the problem and the lack of noninformative priors for experiments when a stopping rule is given, and in part a wish to develop an interesting application of a recently developed method of deriving reference prior by Berger and Bernardo.