Random design wavelet curve smoothing

Antoniadis, Anestis; Grègoire, Gèrard; Vial, Pierre

doi:10.1016/s0167-7152(97)00017-5

Cited by 37 publications

(21 citation statements)

References 5 publications

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“…However, for a deterministic design, one may use several procedures that have been developed to relax these requirements without affecting the results obtained in this paper; such as, for example, the interpolation method of Hall and Turlach [11], the binning method of Antoniadis et al [2], the transformation method of Cai and Brown [5], the isometric method of Sardy et al [24], the interpolation method to a fine regular grid of Kovac and Silverman [17] and the penalized wavelet method of Antoniadis and Fan [1]. In principle, any of these methods can fit into the framework of this paper, with each interpolation method inducing a different choice of a function base for the series expansion.…”

Section: Discussionmentioning

confidence: 99%

Penalized wavelet monotone regression

Antoniadis

Bigot

Gijbels

2007

Statistics & Probability Letters

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In this paper we focus on nonparametric estimation of a constrained regression function using penalized wavelet regression techniques. This results into a convex optimization problem under linear constraints. Necessary and sufficient conditions for existence of a unique solution are discussed. The estimator is easily obtained via the dual formulation of the optimization problem. In particular we investigate a penalized wavelet monotone regression estimator. We establish the rate of convergence of this estimator, and illustrate its finite sample performance via a simulation study. We also compare its performance with that of a recently proposed constrained estimator. An illustration to some real data is given.

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Section: Discussionmentioning

confidence: 99%

Penalized wavelet monotone regression

Antoniadis

Bigot

Gijbels

2007

Statistics & Probability Letters

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“…Several adaptive procedures can be applied for the reconstruction of a signal with unknown smoothness: nonlinear wavelet estimation (thresholding), model selection, kernel estimation with a variable bandwidth (the Lepski method), and so on. Recent results dealing with the adaptive estimation of the regression function when the design is not equispaced or random include Antoniadis et al [1], Baraud [2], Brown and Cai [4], Wong and Zheng [21], Maxim [17], Delouille et al [7], Kerkyacharian and Picard [12], among others.…”

Section: Motivationsmentioning

confidence: 99%

On pointwise adaptive curve estimation based on inhomogeneous data

Gaı̈ffas

2007

ESAIM: PS

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Abstract.We want to recover a signal based on noisy inhomogeneous data (the amount of data can vary strongly on the estimation domain). We model the data using nonparametric regression with random design, and we focus on the estimation of the regression at a fixed point x0 with little, or much data. We propose a method which adapts both to the local amount of data (the design density is unknown) and to the local smoothness of the regression function. The procedure consists of a local polynomial estimator with a Lepski type data-driven bandwidth selector, see for instance Lepski et al. [15]. We assess this procedure in the minimax setup, over a class of function with local smoothness s > 0 of Hölder type. We quantify the amount of data at x0 in terms of a local property on the design density called regular variation, which allows situations with strong variations in the concentration of the observations. Moreover, the optimality of the procedure is proved within this framework.Mathematics Subject Classification. 62G05, 62G08.

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“…Less work has been done for nonequispaced data, the setup we deal with in this article. Classical approaches proposed so far mainly rely on reducing the design to the equispaced case, see the binning methods of Antoniadis, Gregoire, and Vial (1997), and the transformation methods of Cai and Brown (1998). As for Bayesian methods, when the design is nonequispaced inference cannot rely on settings that imply the a posteriori independence of the coefficients, unlike for the case of equispaced data.…”

Section: A Brief Review Of Wavelet Series Expansions For Nonparametrimentioning

confidence: 99%

Bayesian Methods for Wavelet Series in Single-Index Models

Park

Vannucci

Hart

2005

Journal of Computational and Graphical Statistics

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Single-index models have found applications in econometrics and biometrics, where multidimensional regression models are often encountered. This article proposes a nonparametric estimation approach that combines wavelet methods for nonequispaced designs with Bayesian models. We consider a wavelet series expansion of the unknown regression function and set prior distributions for the wavelet coefficients and the other model parameters. To ensure model identifiability, the direction parameter is represented via its polar coordinates. We employ ad hoc hierarchical mixture priors that perform shrinkage on wavelet coefficients and use Markov chain Monte Carlo methods for a posteriori inference. We investigate an independence-type Metropolis-Hastings algorithm to produce samples for the direction parameter. Our method leads to simultaneous estimates of the link function and of the index parameters. We present results on both simulated and real data, where we look at comparisons with other methods.

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Random design wavelet curve smoothing

Cited by 37 publications

References 5 publications

Penalized wavelet monotone regression

Penalized wavelet monotone regression

On pointwise adaptive curve estimation based on inhomogeneous data

Bayesian Methods for Wavelet Series in Single-Index Models

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