Pseudo-true values

Abstract

This preliminary chapter discusses the issue of approximating the distributions of a parametric family F = {F α |α є Ω F ⊂ R m } by those of another parametric family g = {G β |β є Ωg ⊂ R n }. The definition of encompassing will be based on such approximations, which are therefore discussed at some length here. The general problem consists of finding the mapping of F into g which associates with each distribution F α є F the distribution G β є g closest to it according to some criterion. The adoption of the Kullback-Leibler [1951] Information Criterion (KLIC) as a distance measure between distributions defines such a mapping. The mapping which results from this choice will be the sole object of study in this chapter. In line with Sawa [1978], we call the values obtained under this mapping pseudo-true distributions, and their associated parameter vectors pseudo-true (parameter) values. Although only implicitly, pseudo-true values seem to have appeared first in the pioneering work of Cox [1961, 1962] in connection with non-nested hypothesis testing.