Uniform central limit theorems for kernel density estimators

By Evarist Gin'e, Richard Nickl. 2008

85 citations recorded by the source corpus.

Subfields: Management Science and Operations Research, Numerical Analysis, Statistics and Probability

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Citations in the corpus

On the optimal estimation of probability measures in weak and strong topologies (2013, background): “…to P n . k While we showed this fact through Theorems 4 and 5 for the kernel density estimator with F = F H (and Proposition 7 for F = K X ), [16, 17, 15] showed the same result with F being functions of bounded variation, { 1 ( −∞ ,t ] : t ∈ R } , H¨older, Lipschitz and Sobolev classes on R…”
Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions (2018, background): “, as in Lemma 8b in [23]) and we can thus make inference on invariant probabilities μb(C) = ∫ Td 1Cdμ, d = 2, 3.”
Nonparametric Bernstein–von Mises theorems in Gaussian white noise (2012, background): “Standard frequentist estimators such as kernel, wavelet and nonparametric maximum likelihood estimators satisfy this property; in fact, one can even prove a corresponding uniform central limit theorem in l∞(H) for such estimators; see Kiefer and Wolfowitz [24], Nickl [30], Giné and Nickl [21, 22].”
Frequentist coverage and sup-norm convergence rate in Gaussian process regression (2017, methodology): “…kernel representation [32, 38] of the kernel ridge regression estimator to establish a novel local linear expansion of the posterior mean function relative to the supremum norm, which is of independent interest and builds a link between GP regression and frequentist kernel type estimators [21].”
A Donsker Theorem for Lévy Measures (2012, background): “The only differences to Theorem 3 in Giné and Nickl (2008) are in (b), which can be weakened as we only establish tightness in this theorem and not convergence of the finite-dimensional distributions, and in the new condition (d), which replaces translation invariance of F by a more generic conditio”
Inference on counterfactual distributions (2009, unknown): “The requirement Ĝk Gk holds not only for empirical measures but also for various smooth empirical measures; in fact, in the latter case the indexing class of functions F can be much larger than Glivenko-Cantelli or Donsker; see Radulovic and Wegkamp (2003) and Gine and Nickl (2008).”
Adaptation on the space of finite signed measures (2008, methodology): “, Lemma 5b in Giné and Nickl [8], and, with the convention thatDp0 = P , we obtain ‖Dp0 ∗ f̄‖∞ ≤ ‖Dαp0‖TV ‖f‖∞ < ∞, where ‖Dαp0‖TV denotes the total variation norm (see (1) above) of the measure D p0(y)dy, which is equal to theL1-norm ofDp0 forα > 0.”
An exponential inequality for the distribution function of the kernel density estimator, with applications to adaptive estimation (2009, background): “In the recent articles [2,17,18,29], it was shown that several nonparametric density estimators (such as maximum likelihood, wavelet or kernel estimators) are within a ‖ · ‖F -ball of probabilistic size o(1/√n) around the empirical measure, where ‖μ‖F = sup f ∈F | ∫ f dμ| is the usual supnorm of a m”
Global sensitivity analysis for repeated measures studies with informative drop‐out: A semi‐parametric approach (2018, background): “Condition (ii) is standard in the analysis of estimators based on data-adaptive estimation of nuisance parameters—Giné and Nickl (2008) presents conditions under which it is expected to hold.”
A simple adaptive estimator of the integrated square of a density (2008, methodology): “Our 1350-7265 © 2008 ISI/BS main point here consists of an observation on the bias term based on smoothing properties of convolutions, borrowed in part from Giné and Nickl (2007).”
Uniformly Semiparametric Efficient Estimation of Treatment Effects With a Continuous Treatment (2015, background): “Under some assumptions described below, Gine and Nickl (2008), in their Proposition 4, showed that the kernel estimator converges weakly.”
Uniform in Bandwidth Estimation of Integral Functionals of the Density Function (2008, methodology): “The result we obtain generalizes to derivatives of the density and to uniformity in bandwidth a result of Giné & Nickl (2008b) who show, in particular, that, in order to achieve the bias improvement of Bickel & Ritov (1988) over Hall & Marron (1987) for the estimation of the integral of the square…”
Uniform limit theorems for wavelet density estimators (2008, unknown): “Using Lemmas 2 and 3 one can prove the following inequality, which is similar to Theorem 1 in Giné and Nickl (2009). Lemma 4.”