Function-Parametric Empirical Processes, Projections and Unitary Operators

Abstract

We describe another approach to the theory of distribution free testing. The approach uses geometric similarity within various forms of empirical processes: whenever there is an empirical object (like the empirical distribution function) and theoretical parametric model (like a parametric model for distribution function) and a normalised difference of the two, then substitution of estimated values of the parameters leads to projection of this difference. Then one can bring some system in the multitude of these projections. We use unitary operators to describe classes of statistical problems, where one can “rotate” one projection into another, thus creating classes of equivalent problems. As a result, behaviour of various test statistics could be investigated in only one “typical” problem from each class. Thus, the approach promises economy in analytic and numerical work. We also hope to show that the unitary operators involved in “rotations” are of simple and easily implementable form.