An Empirical Process Approach to the Uniform Consistency of Kernel-Type Function Estimators

By U. Einmahl, David Mason. 2000

205 citations recorded by the source corpus.

Subfields: Numerical Analysis, Ocean Engineering, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/17eb1df4ce1a2d46845194a3bc324dccedd57f1c

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Citations in the corpus

Rejoinder: Local Rademacher complexities and oracle inequalities in risk minimization (2006, background): “4) The proofs of these types of bounds can be found in [17, 18, 20, 37, 41]; the current version of (2.”
Asymptotic theory for density ridges (2014, background): “Note that similar conditions to (K2) appear in Einmahl and Mason (2000, 2005), Giné and Guillou (2002).”
Empirical graph Laplacian approximation of Laplace–Beltrami operators: Large sample results (2006, methodology): “The asymptotic properties of empirical laplacians are closely related to the well developed theory of kernel density and kernel regression estimators, which can be viewed as examples of so called local empirical processes, as in [6].”
GLOBAL UNIFORM RISK BOUNDS FOR WAVELET DECONVOLUTION ESTIMATORS (2011, methodology): “, Nolan and Pollard [34], Einmahl and Mason [13], Giné and Guillou [16]), but this does not seem viable in the deconvolution problem, due to the fact that the bounded variation norm does not possess a nice Fourier-analytical characterization.”
An exponential inequality for the distribution function of the kernel density estimator, with applications to adaptive estimation (2009, background): “This was extended to VC classes of functions by several authors ([11,13,15], among others).”
Oracle inequalities in empirical risk minimization and sparse recovery problems (2011, background): “Special cases of the inequalities discussed in Section 3.4 can be found in Talagrand [131], Einmahl and Mason [58], Giné and Guillou [62], Mendelson [108], Giné, Koltchinskii and Wellner [63].”
Nonparametric regression with nonparametrically generated covariates (2012, methodology): “NONPARAMETRICALLY GENERATED COVARIATES 3 a function space with finite integral bracketing entropy, and derives a rate of consistency uniformly over (x, r) ∈ X × Λ; see also Einmahl and Mason (2000) for a related problem.”
On consistency of kernel density estimators for randomly censored data: rates holding uniformly over adaptive intervals (2001, methodology): “(See Einmahl and Mason [8] for a similar exponential bound.”
Uniform in bandwidth consistency of kernel-type function estimators (2005, methodology): “Consult [16] and [11] for references to some of the more important work.”
Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections (2008, background): “Inequalities of this type also have some historical precedents ([9,10,12,32] among others).”
Rates of strong uniform consistency for multivariate kernel density estimators (2002, methodology): “Now we apply Proposition 2 in Einmahl and Mason [6] by following the steps of the proof of their Proposition 3 from Eq.”
On local U-statistic processes and the estimation of densities of functions of several sample variables (2007, methodology): “First we consider the case of [/-processes indexed by VC classes of functions and obtain a moment inequal ity that generalizes the scope of that of Einmahl and Mason [13] and Gine and Koltchinskii [17] for empirical processes.”
Gaussian approximation of suprema of empirical processes (2012, background): “These conditions on the bandwidth hn are mild, and interestingly they essentially coincide with the conditions on the bandwidth used in establishing exact rates of uniform strong consistency of kernel type estimators in [23, 24].”
Nonparametric modal regression (2014, background): “See Gine and Guillou (2002); Einmahl and Mason (2005) for similar findings in density estimation.”