Preservation Theorems for Glivenko-Cantelli and Uniform Glivenko-Cantelli Classes

By A. van der Vaart, Jon A. Wellner. 2000

126 citations recorded by the source corpus.

Subfields: Economics and Econometrics, Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/cc53bce9ed27e432772c5139e743beab0c2767af

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Economics and Econometrics

1. N. Breslow
2. Michael R. Kosorok
3. Jon A. Wellner
4. Jan Peters
5. Rui Song
6. Richard S. Sutton

In Finance

1. N. Breslow
2. Jan Peters
3. Jon A. Wellner
4. Rui Song
5. Michael R. Kosorok
6. G'abor J. Sz'ekely
7. Maria L. Rizzo
8. Arthur Gretton

In Statistics and Probability

1. Michael R. Kosorok
2. N. Breslow
3. Rui Song
4. Jan Peters
5. Pascal Klink
6. Carlo D'Eramo
7. Jon A. Wellner
8. Marcello Restelli

Citations in the corpus

• A Z‐theorem with Estimated Nuisance Parameters and Correction Note for ‘Weighted Likelihood for Semiparametric Models and Two‐phase Stratified Samples, with Application to Cox Regression’ (2008, methodology): “Thus the Glivenko–Cantelli preservation theorem of van der Vaart & Wellner (2000) applies by taking (u, v) = uv, F1 = { ˙0j/2 0}, F2 = ˙ `,, and noting that F1 · F2 has integrable envelope function ˙0j/2 0 F1. A similar argument works”
• FURTHER DETAILS ON INFERENCE UNDER RIGHT CENSORING FOR TRANSFORMATION MODELS WITH A CHANGE-POINT BASED ON A COVARIATE THRESHOLD (2006, background): “If a class of measurable functions F is P -Glivenko-Cantelli with ‖P‖F < ∞, then the class κ · F = {κf : f ∈ F}, where κ denotes a generic version of one of the weights κi, is also P -Glivenko-Cantelli, by theorem 3 of [33].”
• Boosted Fitted Q-Iteration (2017, background): “The preservation theorem (van der Vaart & Wellner, 2000) states that, under mild assumptions, the space obtained by the sum of Glivenko-Cantelli functions is still GlivenkoCantelli.”
• Efficient semiparametric estimation of multi-valued treatment effects under ignorability (2010, unknown): “…β∈B ∣∣M IPW[t],n (β, p∗t )−M IPW[t] (β, p∗t )∣∣ = sup βt∈B ∣∣∣∣∣1n n∑ i=1 Dt,im (Yi;βt) p∗t (Xi) − E [ Dtm (Y ;βt) p∗t (X) ]∣∣∣∣∣ = op (1) because the class of functionsFt = {βt 7→ 1 {· = t}m (·;βt) /p∗t (·) : βt ∈ B} is Glivenko–Cantelli by Assumptions 2(b) and 4 (van der Vaart and Wellner, 2000).”
• Boosted Curriculum Reinforcement Learning (2022, background): “…each weak regressor is Glivenko-Cantelli, i.e., ”the error due to the empirical process goes to zero at least asymptotically”, then, due to the preservation theorem, also the space resulting from the sum of weak regressors is Glivenko-Cantelli (Van Der Vaart & Wellner, 2000; Tosatto et al., 2017).”