Preservation theorems for Glivenko-Cantelli and uniform Glivenko-Cantelli classes

We show that the P-Glivenko property of classes of functions F-1,...,F-k is preserved by a continuous function phi from R-k to R in the sense that the new class of functions x --> phi (f(1)(x),...,f(k)(x)), f(i) is an element of F-i, i = 1,...,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 Y is partitioned into sets which are members of some VC-class C of subsets of Y.