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

Goldenshluger, Alexander; Lepski, Oleg

doi:10.1214/11-aos883

Cited by 188 publications

(244 citation statements)

References 24 publications

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“…The choice of K proposed in in Kerkyacharian et al (2001) and in Goldenshluger and Lepski (2014) implies, see for instance Goldenshluger and Lepski (2011), for any 1…”

Section: Particular Casesmentioning

confidence: 99%

Lower bounds in the convolution structure density model

Lepski¹,

Willer²

2017

Bernoulli

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Abstract:The aim of the paper is to establish asymptotic lower bounds for the minimax risk in two generalized forms of the density deconvolution problem. The observation consists of an independent and identically distributed (i.i.d.) sample of n random vectors in R d . Their common probability distribution function p can be written aswhere f is the unknown function to be estimated, g is a known function, α is a known proportion, and ⋆ denotes the convolution product. The bounds on the risk are established in a very general minimax setting and for moderately ill posed convolutions. Our results show notably that neither the ill-posedness nor the proportion α play any role in the bounds whenever α ∈ [0, 1), and that a particular inconsistency zone appears for some values of the parameters. Moreover, we introduce an additional boundedness condition on f and we show that the inconsistency zone then disappears.AMS 2000 subject classifications: 62G05, 62G20.

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“…The choice of K proposed in in Kerkyacharian et al (2001) and in Goldenshluger and Lepski (2014) implies, see for instance Goldenshluger and Lepski (2011), for any 1…”

Section: Particular Casesmentioning

confidence: 99%

Lower bounds in the convolution structure density model

Lepski¹,

Willer²

2017

Bernoulli

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“…Our selection method of the dimension parameter is inspired by the work of Goldenshluger and Lepski [2011] and combines the techniques of model selection and Lepski's method. We determine the dimension parameter among a collection of admissible values by minimizing a penalized contrast function.…”

Section: Methodology and Backgroundmentioning

confidence: 99%

“…, Z n from a minimax point of view. The estimator is based on an orthogonal series approach where the fully data-driven selection of the dimension parameter is inspired by the recent work of Goldenshluger and Lepski [2011]. We derive conditions that allow us to bound the maximal risk of the fully data-driven estimator over suitable chosen classes F for f , which are constructed flexibly enough to characterize, in particular, differentiable or analytic functions.…”

Section: Introductionmentioning

confidence: 99%

Adaptive nonparametric estimation in the presence of dependence

Asin

Johannes

2017

Journal of Nonparametric Statistics

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We consider non-parametric estimation problems in the presence of dependent data, notably non-parametric regression with random design and non-parametric density estimation. The proposed estimation procedure is based on a dimension reduction. The minimax optimal rate of convergence of the estimator is derived assuming a sufficiently weak dependence characterized by fast decreasing mixing coefficients. We illustrate these results by considering classical smoothness assumptions. However, the proposed estimator requires an optimal choice of a dimension parameter depending on certain characteristics of the function of interest, which are not known in practice. The main issue addressed in our work is an adaptive choice of this dimension parameter combining model selection and Lepski's method. It is inspired by the recent work of Goldenshluger and Lepski [2011]. We show that this data-driven estimator can attain the lower risk bound up to a constant provided a fast decay of the mixing coefficients.

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“…Data-driven methods exist, such as the one very recently introduced by Goldenshluger and Lepski [25,24,17], which permit one to choose the optimal parameter h.…”

Section: Kernel Density Estimationmentioning

confidence: 99%

Size Distribution of Amyloid Fibrils. Mathematical Models and Experimental Data

Prigent¹,

Haffaf²,

Banks³

et al. 2014

Int. J. of Pure and Appl. Math.

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More than twenty types of proteins can adopt misfolded conformations, which can co-aggregate into amyloid fibrils, and are related to pathologies such as Alzheimer's disease. This article surveys mathematical models for aggregation chain reactions, and discuss the ability to use them to understand amyloid distributions. Numerous reactions have been proposed to play a role in their aggregation kinetics, though the relative importance of each reaction in vivo is unclear: these include activation steps, with nucleation compared to initiation, disaggregation steps, with depolymerization compared to fragmentation, and additional processes such as filament coalescence or secondary nucleation. We have statistically analysed the shape of the size distribution of prion fibrils, with the specific example of truncated data due to the experimental technique (electron microscopy). A model of polymerization and depolymerization succeeds in explaining this distribution. It is a very plausible scheme though, as evidenced in the review of other mathematical models, other types of reactions could also give rise to the same type of distributions.

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

Cited by 188 publications

References 24 publications

Lower bounds in the convolution structure density model

Lower bounds in the convolution structure density model

Adaptive nonparametric estimation in the presence of dependence

Size Distribution of Amyloid Fibrils. Mathematical Models and Experimental Data

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