Empirical processes indexed by estimated functions

By A. van der Vaart, Jon A. Wellner. 2007

132 citations recorded by the source corpus.

We consider the convergence of empirical processes indexed by functions that depend on an estimated parameterand give several alterna- tive conditions under which the "estimated parameter" �n can be replaced by its natural limit �0 uniformly in some other indexing set �. In particular we reconsider some examples treated by Ghoudi and Remillard (Asymptotic Methods in Probability and Statistics (1998) 171-197, Fields Inst. Commun. 44 (2004) 381-406). We recast their examples in terms of empirical process theory, and provide an alternative general view which should be of wide applicability.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/79fea67486636bb2d8816343680c0ace4135d9e9

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In Artificial Intelligence

In Statistics and Probability

Citations in the corpus

Uniformly Semiparametric Efficient Estimation of Treatment Effects With a Continuous Treatment (2015, unknown): “As for condition G.5, by Corollary 9.32 (iii) of Kosorok (2008), condition G.V implies that {ψβ,π,t : β ∈ ∞δ (T ), π ∈ δ, t ∈ T } is Donsker, which in turn implies G.5’ by Lemma 3.3.5 of van der Vaart and Wellner (1996).”
Valid Post-Selection Inference in High-Dimensional Approximately Sparse Quantile Regression Models (2013, background): “7 in [39] 1 2 P (|En[ruizui]| > t) 6 2P (|En[ruizuiǫi]| > t/4) Since the ruizuiǫi is a symmetric random variable and |zui| 6 1, by Theorem 2.”
Large‐sample tests of extreme‐value dependence for multivariate copulas (2011, unknown): “If C[j](u) exists and is continuous on [0,1]d for all j∈{1, . . . , d}, then, from Corollary 5.3 of van der Vaart & Wellner (2007) (see also Stute, 1984; Gänssler & Stute, 1987; Fermanian, Radulovic, &Wegkamp 2004; Tsukahara, 2005), the empirical copula process Cn = √n(Cn−C) converges weakly in…”
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): “As argued in van der Vaart & Wellner (2007, p. 235) , we can derive limit distributions of ˆn based on {n( ,ˆ ): ∈} from the corresponding theory for {n(,0 ): ∈}, if we know that √ n(ˆn −0) = Op(1) and if we show that”