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
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.
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van der Vaart, A., Wellner, J.A. (2000). Preservation Theorems for Glivenko-Cantelli and Uniform Glivenko-Cantelli Classes. In: Giné, E., Mason, D.M., Wellner, J.A. (eds) High Dimensional Probability II. Progress in Probability, vol 47. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1358-1_9
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DOI: https://doi.org/10.1007/978-1-4612-1358-1_9
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