Reconciling frequentist properties with the likelihood principle

Abstract

For the general problem of parametric statistical inference, several frequentist principles are formulated, including principles of hypothesis testing, set estimation, and conditional inference. These principles guarantee that, whatever the true parameter value, statistical procedures have little chance of producing misleading inferences. The frequentist principles are shown to be compatible with the likelihood principle and with principles of coherence. Two general methods are studied which satisfy both the likelihood and frequentist principles in finite samples. One method produces posterior upper and lower probabilities from a very large set of prior probability measures, which can be taken to be an ε-contamination neighborhood with ε slightly larger than 1 2 . The second method derives inferences from a normalized version of the observed likelihood function. Because inferences from the two methods encompass a wide range of frequentist, likelihood and Bayesian inferences, they are conservative and they have relatively low power. More powerful methods can be obtained by weakening the frequentist principles and making weak assumptions about the sampling rule. The results show that there are methods of statistical inference, based on particular types of imprecise probability model, which satisfy the likelihood principle, are coherent, and have good frequentist properties under a range of sampling models.