On the foundations of statistics: A frequentist approach

Frank Hampel

June 1998

Abstract: A limited but basic problem in the foundations ofstatistics is the following: Given a parametric model, given perhaps someobservations from the model, but given no prior information about theparameters ("total ignorance"), what can we say about the occurrence of aspecified event A under this model in the future (prediction problem)? Or,as probabilities are often described in terms of bets, how can we bet on A?Bayesian solutions are internally consistent and fully conditional onthe observed data, but their ties to the observed reality and theirfrequentist properties can be arbitrarily bad (unless, of course, theassumed prior distribution happens to be the true prior). Frequentistsolutions are generally not possible with ordinary probabilities; but it ispossible to define "successful bets" (using upper and lower probabilities),which even lead out of the state of total ignorance in an objectivelearning process converging to the true probability model. A specialvariant (successful bets on random parameter sets) provides a new andcorrect interpretation of the basic idea of Fisher's "fiducialprobabilities." Successful bets (which are only one-sided and not fully conditional) canbe used for inference, but not for decisions which, as in Bayesian "fairbets," require ordinary probabilities. However, it is possible to define"best enforced fair bets" (or corresponding probability distributions)which solve the decision problem in a specific minimax sense. In as much asthey are also Bayes solutions, they may be called "least unsuccessful Bayessolutions" (providing another candidate for "least informative priors"). Onthe other hand, among the successful bets we can select the "least unfairsuccessful bets" which in a way come closest to Bayes solutions. Several(nontrivial) examples, mainly for two independent binomials, have alreadybeen worked out by R. Steiner and the author.

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