Empirical Processes in M-Estimation

By Sara A. van de Geer. 2000

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Additive models with trend filtering (2017, background): “As for part (ii), we note that Lemma 4 of Wang et al. [2014] shows that |hj`(X j)− gj`(X j)| ≤ k(2)Wj , for ` = k + 2, .”
Optimal Rates for Random Fourier Features (2015, methodology): “[10] D.”
On the empirical estimation of integral probability metrics (2012, neutral): “7 of [49], supf∈F |Pmf−Pf | a.”
Variance-based Regularization with Convex Objectives (2016, methodology): “ +2 1 +2 r 2t n ! ǫ. This gives the theorem. D Proof of Theorem 4 We ﬁrst show the following version of uniform Bernstein’s inequality with Rademacher complexities. The proof uses a peeling technique [3, 46], in conjuction with Talagrand’s concentration inequality (Lemma B.2). Lemma D.1. Let r > ”
Estimating Divergence Functionals and the Likelihood Ratio by Convex Risk Minimization (2008, methodology): “stein distance. This is deﬁned as follows: for a constant K>0, ρK(f)2 := 2K2 Z (e|f|/K −1 −|f|/K)dP. The Bernstein distance is related to the Hellinger distance in several crucial ways (see, e.g., [26], page 97): • ρ1(fg) ≤ 4hQ(g0, g+g0 2 ). • The bracketing entropy based on Bernstein distance is”
On the mathematical foundations of learning (2001, background): “On the other hand, entropy numbers have a strong history in related contexts (see [21], [44], [24], [41]).”
CAM: Causal Additive Models, high-dimensional order search and penalized regression (2013, background): “see Bühlmann and van de Geer (2011, Ch.5.6,Th.8.2). Assumption (B2,ii) can be ensured by assuming maxj |Aj | ≤M1 <∞ and e.g. group restricted eigenvalue assumptions for the design matrix (with the given basis); see Zhang and Huang (2008) and van de Geer et al.”
Local Rademacher complexities (2005, methodology): “Results of this type have been obtained, among others by van de Geer [33, 34] who even provides non-asymptotic exponential inequalities.”
On integral probability metrics, φ-divergences and binary classification (2009, positive): “The following result is quoted from [52, Theorem 3.7].”
On strong identifiability and convergence rates of parameter estimation in finite mixtures (2016, methodology): “If we fit the data to a mixture of k = k0+1 = 4 Gaussian distributions, however, we observe empirically that the convergence rate of W4(Ĝn, G0) (thus W2 distance) is almost n−1/8 up to a logarithmic term.”
Obfuscation via Information Density Estimation (2019, background): “(27) Moreover, since G is compact and g is continuous, the functions g and eg satisfy the uniform law of large numbers [60].”
Robust semiparametric M-estimation and the weighted bootstrap (2005, methodology): “This aspect has been well studied for general M-estimators in VW and van de Geer (2000).”
Threshold quantile autoregressive models (2011, neutral): “Therefore by Lemma 3.10 in van de Geer (2000) we have that H1,B(δ,P, P ) ∞, that is, the δ-entropy with bracketing of P is finite.”
Statistical Analysis of Some Multi-Category Large Margin Classification Methods (2004, methodology): “This method, widely studied in the statistics literature, is often referred to ashe method of sieves(for example, see Chapter 10 of van de Geer, 2000, and references therein).”
Regularized Policy Iteration with Nonparametric Function Spaces (2016, methodology): “2 by van de Geer (2000), with two main differences: 1) It holds uniformly over Q and 2) ĥn(·;Q) uses the same data Dn that is used to estimate Q itself.”