The Conditional Predictive Ordinate for the Normal Distribution

Abstract

SUMMARY The conditional predictive ordinate (CPO) is a Bayesian diagnostic which detects surprising observations. It has been used in a variety of situations such as univariate samples, the multivariate normal distribution and regression models. Results are presented about the most surprising observation which has minimum CPO. For the multivariate normal distribution it is shown that the most surprising observation must lie at one of the vertices of the convex hull. It is also shown that the observation with maximum Mahalanobis distance from the sample mean must lie on the convex hull. Results are given for the expected number of vertices on the convex hull when the sample is contaminated. An alternative, closely related diagnostic, the ratio ordinate measure, is presented. A numerical comparison of the two measures is given.