Preservation Theorems for Glivenko-Cantelli and Uniform Glivenko-Cantelli Classes
نویسنده
چکیده
We show that the P−Glivenko property of classes of functions F1, . . . ,Fk is preserved by a continuous function φ from R to R in the sense that the new class of functions x → φ(f1(x), . . . , fk(x)), fi ∈ Fi, 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 (1998). 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.
منابع مشابه
Preservation Theorems for Glivenko-cantelli and Uniform Glivenko-cantelli Classes Aad Van Der Vaart and Jon
We show that the P Glivenko property of classes of functions F1; : : : ;Fk is preserved by a continuous function ' from R k to R in the sense that the new class of functions x! '(f1(x); : : : ; fk(x)); fi 2 Fi; 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 pro...
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