Stochastic orders and comparison of experiments

Abstract

Exploring criteria for majorization, exact and approximate, univariate and multivariate, we relate them to criteria for information orderings of statistical experiments. After providing some basic criteria for comparison pf experiments, we observe their straightforward generalizations to general families of measures. Thus LeCam's randomization criterion extends to a criterion for comparing families of measures. Reversing the randomizations, we obtain dilation like kernels mapping densities, exactly or approximately, into densities. Using this, we derive criteria for comparison of measures in terms of integrals of given functions. In particular we obtain well-known criteria for one measure being a dilation of another measure and for stochastic orderings of distributions on partially ordered sets. Experiments having two point parameters sets, i.e. dichotomies, enjoy a variety of striking properties which are not shared by experiments in general. Dichotomies may be studied in terms of their Neyman-Pearson functions, which are functions describing the relationships between the probabilities of errors of the two kinds for most powerful tests. These functions are the inverses of the Lorenz functions of econometrics. Observing this, we readily obtain various criteria for one distribution being approximately Lorenz majorized by another.