Preservation Theorems for Glivenko-Cantelli and Uniform Glivenko-Cantelli Classes

Abstract

We show that the P—Glivenko property of classes of functions F ₁,…,F _k is preserved by a continuous function ϕ from R ^k to R in the sense that the new class of functions

$$x \to \varphi ({f_1}(x), \ldots ,{f_k}(x)),{f_i} \in {F_i},i = 1, \ldots ,k$$

is again a Glivenko-Cantelli class of functions if it has an integrable envelope. We also prove an analogous result for preservation of the uniform Glivenko-Cantelli property. Corollaries of the main theorem include two preservation theorems of Dudley (1998a,b). We apply the main result to reprove a theorem of Schick and Yu (1999) concerning consistency of the NPMLE in a model for “mixed case” interval censoring Finally a version of the consistency result of Schick and Yu (1999) is established for a general model for “mixed case interval censoring” in which a general sample space У is partitioned into sets which are members of some VC-class C of subsets of У.