Preservation Theorems for Glivenko-cantelli and Uniform Glivenko-cantelli Classes Aad Van Der Vaart and Jon
نویسنده
چکیده
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 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. 1. Glivenko Cantelli Theorems. Let (X ;A; P ) be a probability space, and suppose that F L1(P ). For such a class of functions, let F0;P ff Pf : f 2 Fg. We also let FF (x) supf2F jf(x)j, the envelope function of F . If jf j F for all f 2 F with F measurable, then F is an envelope for F . Suppose that X1; X2; : : : are i.i.d. P . Glivenko-Cantelli theorems give conditions under which the empirical measure Pn converges uniformly to P over a class F , either in probability (in which case we say that F is a weak Glivenko-Cantelli class for P ) or almost surely: kPn PkF sup f2F jPnf Pf j !a:s: 0 ; in this case we say that we say that F is a strong Glivenko-Cantelli class for P . Useful suÆcient conditions for a class F to be a strong Glivenko-Cantelli class for P are that it has an integrable envelope and either logN[]( ;F ; L1(P )) <1 for all > 0
منابع مشابه
Preservation Theorems for Glivenko-Cantelli and Uniform Glivenko-Cantelli Classes
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 prop...
متن کاملSome Converse Limit Theorems for Exchangeable Bootstraps
The bootstrap Glivenko-Cantelli and bootstrap Donsker theorems of Giné and Zinn (1990) contain both necessary and sufficient conditions for the asymptotic validity of Efron’s nonparametric bootstrap. In the more general case of exchangeably weighted bootstraps, Praestgaard and Wellner (1993) and Van der Vaart and Wellner (1996) give analogues of the sufficiency half of the Theorems of Giné and ...
متن کاملUniform Glivenko–Cantelli Classes
A class of sets, or functions, is said to be P–Glivenko–Cantelli if the empirical measure Pn converges in some sense to the true measure, P , as n → ∞, uniformly over the class of sets or functions. Thus, the notions of Glivenko–Cantelli, and likewise uniform Glivenko–Cantelli are for the most part qualitative assessments of how “well–behaved” a collection of sets or functions is, in the sense ...
متن کاملA Counterexample Concerning the Extension of Uniform Strong Laws to Ergodic Processes
We present a construction showing that a class of sets C that is Glivenko-Cantelli for an i.i.d. process need not be Glivenko-Cantelli for every stationary ergodic process with the same one dimensional marginal distribution. This result provides a counterpoint to recent work extending uniform strong laws to ergodic processes, and a recent characterization of universal Glivenko Cantelli classes.
متن کامل