The problem of regions

In the problem of regions, we wish to know which one of a discrete set of p ossibilities applies to a continuous parameter vector. This problem arises in the following way: we compute a descriptive statistic from a set of data , notice an interesting feature and wish to assign a confidence level to th at feature. For example, we compute a density estimate and notice that the estimate is bimodal. What confidence can we assign to bimodality? A natural way to measure confidence is via the bootstrap: we compute our descriptive statistic on a large number of bootstrap data sets and record the proporti on of times that the feature appears. This seems like a plausible measure o f confidence for the feature. The paper studies the construction of such co nfidence values and examines to what extent they approximate frequentist p- values and Bayesian a posteriori probabilities. We derive more accurate con fidence levels using both frequentist and objective Bayesian approaches. Th e methods are illustrated with a number of examples, including polynomial m odel selection and estimating the number of modes of a density.