Uniform in bandwidth consistency of kernel-type function estimators

By U. Einmahl, David Mason. 2005

308 citations recorded by the source corpus.

We introduce a general method to prove uniform in bandwidth consistency of kernel-type function estimators. Examples include the kernel density estimator, the Nadaraya-Watson regression estimator and the conditional empirical process. Our results may be useful to establish uniform consistency of data-driven bandwidth kernel-type function estimators.

Subfields: Artificial Intelligence, Control and Systems Engineering, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/84ea372258703eff0e2d4999a188961752a6d6f1

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In Artificial Intelligence

In Control and Systems Engineering

In Statistics and Probability

Citations in the corpus

UNIFORM BIAS STUDY AND BAHADUR REPRESENTATION FOR LOCAL POLYNOMIAL ESTIMATORS OF THE CONDITIONAL QUANTILE FUNCTION (2011, background): “See in particular Tsybakov (1986) for optimal pointwise consistency rates, Fan (1992) for design adaptation, and Fan and Gijbels (1996) and Loader (1999) for a general overview. The present paper is perhaps more specifically related to Truong (1989), Chauduri (1991), Holderlein and Mammen (2009), an”
Asymptotic normality of plug-in level set estimates. (2009, background): “Theorem 1 of Einmahl and Mason (2005) implies that when hn satisfies (H) and f is bounded that for some constant γ1 > 0 lim sup n→∞ √ nhn logn sup x∈Rd |fn(x) − Efn(x)| ≤ γ1, a.s.(2.15) It is not difficult to see that under the assumptions (D), (K) and (H) for some γ2 > 0, sup n≥2 √ nhn sup x∈Rd…”
Consistency of the kernel density estimator: a survey (2012, background): “The research by Einmahl and Mason was only possible because of earlier research by M. Talagrand dealing with bounds for the supremum of empirical processes. He chose a more general approach than Stute, who restricted himself to the empirical process for the set of functions gx(·) = I(−∞,x](·) (index”
Moving Beyond Sub-Gaussianity in High-Dimensional Statistics: Applications in Covariance Estimation and Linear Regression (2018, methodology): “This arises in uniform-in-bandwidth results related to linear kernel averages; see Theorem 1 of Einmahl and Mason (2005) for a precise problem.”
Uniform limit theorems for wavelet density estimators (2008, unknown): “…stated form because Talagrand’s inequality can be proved [e.g., Ledoux (2001), page 144ff] by estimation of the Laplace transform of ‖∑ni=1 f (Xi)‖F , and exp{λ‖∑ki=1 f (Xi)‖F }, k = 1,2, . . . , is a submartingale, so that Doob’s inequality can be applied [see also Einmahl and Mason (2000, 2005)].”
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 [26, 27].”
A tutorial on kernel density estimation and recent advances (2017, methodology): “According to empirical process theory [10–13], the uniform error sup x jbpn x ð Þ p x ð Þj 1⁄4 O h(2) þ OP ffiffiffiffiffiffiffiffiffi logn nhd r ! (3) under mild conditions (see [11,12] for more details).”
Kernel-Based Semiparametric Estimators: Small Bandwidth Asymptotics and Bootstrap Consistency (2018, background): “It would be of interest to develop uniform versions of Theorems 1 and 2 along the lines of Romano and Shaikh (2012) and Einmahl and Mason (2005), respectively.”
Nonparametric modal regression (2014, methodology): “The first assertion can be proved by the same method in Einmahl and Mason (2000, 2005); Gine and Guillou (2002) and the second assertion is an application of the Talagrand’s inequality (Talagrand, 1996).”