Some Uses of Order Statistics in Bayesian Analysis
by Seymour Geisser
Technical Report No. 604
School of Statistics
University of Minnesota
February, 1995
Research supported in part by grant NIGMS 25271.
Introduction
Order statistics, although playing a prominent role in frequentist methodology,
especially in nonparametric inference, are not often featured in Bayesian
analysis. One area, however, where order statistics can be of interest to
Bayesians is in the detection of outliers or discordant observations. These
situations are such that there is an observation that appears to be somewhat
removed from the remaining ones but no discernible alternative can readily be
specified for the potential discordancy. Alternatives, although sometimes
considered, Dixit (1994), require additional distributional assumptions and
prior probabilities over and above the original model assumptions that
initially an investigator may not be prepared to contemplate. In these
situations a simple Bayesian test of significance may be appropriate
in determining whether an observation or several of them are discordant or
if it is necessary to contemplate alternative model assumptions for the
entire data set. Sections 2 through 6 will consider Bayesian discordancy
testing. Another area involves situations where the calculation of the
probability that the minimum (maximum) of a set of future observables is
greater (smaller) than some critical threshold. More generally we shall
be interested in the chance that