Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality

By Alexander Goldenshluger, O. Lepski. 2010

261 citations recorded by the source corpus.

We address the problem of density estimation with L s-loss by selection of kernel estimators. We develop a selection procedure and derive corresponding L s-risk oracle inequalities. It is shown that the proposed selection rule leads to the estimator being minimax adaptive over a scale of the anisotropic Nikol'skii classes. The main technical tools used in our derivations are uniform bounds on the L s-norms of empirical processes developed recently by Goldenshluger and Lepski [Ann. Probab. (2011), to appear].

View Paper: https://www.semanticscholar.org/paper/61b371f1b26af6f2503f7ff27974420658af1f26

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Citations in the corpus

Functional linear regression with functional response (2017, methodology): “This model is discussed in the monographs by Ramsay and Silverman (2005), Ferraty and Vieu (2006), and Horvath and Kokoszka (2012).”
Adaptive functional linear regression (2011, methodology): “ection method combines model selection and Lepski’s method (cf. [27] and its recent review in [30]) which is inspired by a bandwidth selection method in kernel density estimation proposed recently in [18]. Selecting the dimension parameter mbas minimizer of a stochastic penalized contrast function i”
On adaptive minimax density estimation on $$R^d$$Rd (2012, methodology): “The representation (2.2)–(2.3) is obtained by a straightforward change of variables in the convolution integral [see the proof of Lemma 12 in Goldenshluger and Lepski (2011a)].”
A tutorial on kernel density estimation and recent advances (2017, methodology): “In addition to the above approaches, Goldenshluger and Lepski (2011) propose a method, known as the Lepski’s approach (Goldenshluger and Lepski, 2008; Lepski and Goldenshluger, 2009), that treats the bandwidth selection problem as a model selection problem and proposes a new criterion for selecting ”
Hypothesis Testing for High-Dimensional Multinomials: A Selective Review (2017, background): “Other works developing a local perspective in testing and estimation include Donoho and Johnstone (1994), Goldenshluger and Lepski (2011), Cai and Low (2015), Chatterjee, Guntuboyina and Sen (2015), Wei and Wainwright (2017).”