Applications of empirical process theory

By Sara A. van de Geer. 2000

272 citations recorded by the source corpus.

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Citations in the corpus

Fast learning rates for plug-in classifiers (2005, background): “Indeed, if η is of bounded variation, there exist estimators of η converging with the mean squared L 2 rate n − 2 / 3 (cf. [9, 16, 23, 27]), and thus application of (5.4) immediately yields the rate n − 2(1+ α ) / (3(2+ α )) for the corresponding plug-in rule.”
Convergence of algorithms for reconstructing convex bodies and directional measures (2006, methodology): “The following result is implicit in pages 187–188 of [32].”
Convergence of functional k-nearest neighbor regression estimate with functional responses (2011, background): “9 in [21] (Berstein’s inequality for martingales), we obtain the desired bound.”
Convergence rates for the generalized Fréchet mean via the quadruple inequality (2018, background): “orollary 2 Deﬁne ∆ n(δ) := sup q∈Q: l(m,q)≤δ F(q) −F(m) −F n(q) + F n(m). Results similar to following Lemma are well known in the M-estimation literature. The proof relies on the peeling device, see [Gee00]. Lemma 2 (Weak argmin transform). Assume Growth. Let ζ ≥1. Assume that there are constants ξ”
Consistent covariate selection and post model selection inference in semiparametric regression (2004, methodology): “…normal estimators ofβ, provided that the estimators off satisfy the minimum requirement ‖f − f̂ ‖2µT = oP (n−1/2),(1.2) for α > 1/2, where ‖ · ‖µT is the L2(µT ) norm andµT is the probability distribution of T ; see, for instance, Lemma 11.2, page 202, in van der Geer (2000), or Chen (1988).”
Uniform Inference for High-dimensional Quantile Regression: Linear Functionals and Regression Rank Scores (2017, background): “Such a cover can be constructed with Nδ ≤ ( p t ) (3/ξn) t (see, for example, Lemma 2.5 in van de Geer, 2000).”
SIN: Semantic Inference Network for Few-Shot Streaming Label Learning (2022, background): “Proof: This can be achieved by the uniform concentration bounds developed in empirical process [69], [70] and statistical learning theory [71]–[73].”
Isotonic regression in general dimensions (2017, background): “As rates obtained from the entropy integral (e.g. van de Geer, 2000, Theorem 9.1) do not match those from Sudakov lower bounds for such classes, standard entropy methods result in a non-trivial gap between the minimax rates of convergence, which typically match the Sudakov lower bounds (e.g. Yang…”
PAC-BAYESIAN SUPERVISED CLASSIFICATION: The Thermodynamics of Statistical Learning (2007, methodology): “It can be seen as some variant of the more classical approach of M -estimators relying on empirical process theory — as described for instance in Van de Geer (2000).”
A modern introduction to probability and statistics (2026, background): “, [36]), one can show that asymptotically (that is, as the size n of the dataset goes to infinity) maximum likelihood estimators are unbiased.”
Adaptive confidence sets in $$L^2$$ (2011, unknown): “, [33, 34]).”