Preservation theorems for Glivenko-Cantelli and uniform Glivenko-Cantelli classes. (English) Zbl 0967.60037

Giné, Evarist (ed.) et al., High dimensional probability II. 2nd international conference, Univ. of Washington, DC, USA, August 1-6, 1999. Boston, MA: Birkhäuser. Prog. Probab. 47, 115-133 (2000).
Main result. Suppose that \({\mathcal F}_1, \dots, {\mathcal F}_k\) are \(P\)-Glivenko-Cantelli classes of functions on a probability space \((\Omega, {\mathcal A},P)\) and that \(\varphi: \mathbb{R}^k\to \mathbb{R}\) is continuous. Then the new class of functions \(\varphi({\mathcal F}_1, \dots, {\mathcal F}_k)\) is again Glivenko-Cantelli provided that it has an integrable envelope function. The authors give applications of this result to consistency of the nonparametric maximum likelihood estimation in a model for “mixed case” interval censoring.
For the entire collection see [Zbl 0948.00040].


60F15 Strong limit theorems
62G05 Nonparametric estimation