Weak Convergence and Empirical Processes: With Applications to Statistics

By Thomas Mikosch, A. van der Vaart, Jon A. Wellner. 1996

Subfields: Statistics and Probability, Statistical Learning Theory

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Citations in the corpus

Certifying Some Distributional Robustness with Principled Adversarial Training (2017, methodology): “Thus, the functional θ 7→ F n ( θ ) satisﬁes bounded diﬀerences [14, Thm. 6.2], and applying standard results on Rademacher complexity [1] and entropy integrals [64, Ch. 2.2] gives the result.”
Convex-constrained Sparse Additive Modeling and Its Extensions (2017, background): “…classes M j and C j , their “sizes” are quite diﬀerent: the metric entropy of M j (under the ∞ distance) is known to be of the order (cid:15) − 1 [vdVW00, Theorem 2.7.5], while the metric entropy of the class of convex and uniformly bounded functions is of the signiﬁcantly smaller order (cid:15)…”
Interpretable Distribution Features with Maximum Testing Power (2016, background): “For a class of kernels K on X ⊆ R d , deﬁne (2) Assume that (1) K is a uniformly bounded (by and (3) and c 3) be the universal constants associated to F i -s according to Theorem 2.6.7 in van der Vaart and Wellner (2000).”
Variance-based Regularization with Convex Objectives (2016, methodology): “As a motivating example, we give the following standard bound on the covering number of Lipschitz losses [24].”
Estimating Divergence Functionals and the Likelihood Ratio by Convex Risk Minimization (2008, methodology): “. (13) 3.1 Consistency of estimator E1 Our analysis of consistency relies on tools from empirical process theory. Let us brieﬂy review the notion of the metric entropy of function classes (see, e.g., [28] for further background). For any r≥ 1 and distribution function Q, deﬁne the empirical Lr metri”
Local Rademacher complexities (2005, background): “Unfortunately,one ofthe shortcom ingsofthe Radem acheraveragesisthatthey provide global estim atesofthecom plexity ofthefunction class,thatis,they do notre ect thefactthatthealgorithm willlikely pick functionsthathavea sm allerror, and in particular,only a sm allsubsetofthe function classwillbe…”
On integral probability metrics, φ-divergences and binary classification (2009, methodology): “rgences. Next, in Section III-B, we show that these estimators are strongly consistent and provide their rates of convergence, using concentration inequalities and tools from empirical process theory [20]. In Section III-C, we describe simulation results that demonstrate the practical viability of t”
High-dimensional graphs and variable selection with the Lasso (2006, neutral): “11 in [17]), and λ ∼ dn−(1−ε)/2 with ε > 0, there exists for every g > 0 some c > 0 so that P(|2n−1〈Va,W⊥b 〉| ≥ gλ) ≤ P(|2n−1〈Va,Wb〉| ≥ gλ) = O(exp(−cnε)).”
Empirical margin distributions and bounding the generalization error of combined classifiers (2002, methodology): “One can ﬁnd in the literature (see, e.g., van der Vaart and Wellner (1996)) various upper bounds on such quantities as G n ( ) and R n ( ) in terms of entropies, VC-dimensions, etc.”
Robustness and generalization (2010, methodology): “We recall the following standard notion of covering number from van der Vaart and Wellner (2000). Definition 1 (cf.”