Block thresholding and wavelet estimation for nonequispaced samples

doi:10.1016/S0378-3758(02)00238-0

Journal of Statistical Planning and Inference

Volume 116, Issue 1, 1 September 2003, Pages 113-129

https://doi.org/10.1016/S0378-3758(02)00238-0 Get rights and content

Abstract

For samples with the design points occurring as a Poisson process or having a uniform distribution, the wavelet method of block thresholding can be applied directly to the data as though it was equispaced without sacrificing adaptivity or optimality. When the underlying true function is in certain Besov and Hölder classes, the resulting estimator achieves the minimax rate of convergence. Simulation results are examined.

Introduction

Wavelets have been shown to be very successful in nonparametric function estimation. Specifically, they excel in the areas of spatial adaptivity, optimality, and low computational cost. Typically, wavelet analysis is performed through the use of term-by-term thresholding of wavelet coefficients, such as the VisuShrink method of Donoho and Johnstone (1994). There, a noisy signal is transformed into empirical wavelet coefficients by the discrete wavelet transform, these coefficients are denoised by comparison with a specified thresholding rule, and the underlying function is estimated by applying the inverse discrete wavelet transform to these denoised coefficients. This method is adaptive and is within a logarithmic factor of the optimal minimax convergence rate over large classes of Besov functions.

Hall et al. (1999) proposed a method of wavelet analysis whereby the optimal minimax convergence rate is attained without the logarithm penalty found in the term-by-term methods. Using block thresholding, where the empirical wavelet coefficients are thresholded in blocks rather than individually, they achieved this optimal rate and maintained adaptivity over a large class of Hölder functions. By looking at coefficients in blocks rather than individually, more precise comparisons between the coefficients and the threshold is allowed, resulting in improved rates. Cai (1999) has extended this idea. Using a James–Stein thresholding rule, he has shown that this block-thresholded wavelet estimator attains the optimal convergence rate from both a global and local estimation perspective. A specific block length and threshold value are established that attain these rates. Additionally, this estimate is easy to implement and has a low computational cost of O(n).

The above methods have been developed for data that is equispaced. Little emphasis has been placed on sample data that is not equispaced, however. Cai and Brown (1998) investigated wavelet methods on samples with fixed, nonequispaced designs via an approximation approach. They showed that applying the methods devised for equispaced data directly to nonequispaced data can lead to suboptimal estimators. They then proposed a method that was adaptive and near optimal. Hall and Turlach (1997) used interpolation methods to deal with samples with random design. Unfortunately, these methods are much more complex from a computational standpoint than their equispaced counterparts.

Cai and Brown (1999) have also examined convergence rates when the unknown function is in a Hölder class with exponent α and the positions of the sample points are distributed as independent uniform random variables. Using term-by-term thresholding, they showed that the equispaced wavelet method can be directly applied to the nonequispaced data without a loss in the rate of convergence, i.e., to within a logarithmic factor of the optimal convergence rate of n^{−2α/(2α+1)}. This method maintains the computational efficiency and simplicity of the equispaced algorithm.

In this paper, it is known that these results of Cai and Brown (1999) for uniform design can be improved upon for Hölder classes and extended to many Besov classes through the use of block thresholding. Additionally, when the sample points occur as a Poisson process, the same optimal rates are attained. Adaptivity is kept, and the computational cost remains low since the equispaced algorithm is used. In this paper, the rates given are upper bounds on the rate of convergence. For a discussion on the lower bound of the rate of convergence, see Antos et al. (2000).

In Section 2 of this paper, the method of thresholding the wavelet coefficient is stated, and the theorems on convergence rates are put forth. Section 3 discusses simulation results, Section 4 defines the functions spaces of interest, and Section 5 contains the proofs.

Section snippets

Wavelets

φ and ψ will represent the father and mother wavelets, respectively. Both are assumed to be compactly supported. Let φ_jk and ψ_jk be the translations and dilations of φ and ψ: $φ_{jk} (x)=2^{j/2} φ(2^{j} x−k),$ $ψ_{jk} (x)=2^{j/2} ψ(2^{j} x−k).$ In this paper, wavelets periodized to the interval [0,1] will be used. The set ${φ_{j_{0}k}^{p} : k=1,…,2^{j_{0}}}∪{ψ_{jk}^{p} : j⩾j_{0}, k=1,…,2^{j}}$ is an orthonormal basis of L²[0,1], where $φ_{jk}^{p} (x)= ∑ l=−∞ ∞ φ_{jk} (x−l),$ $ψ_{jk}^{p} (x)= ∑ l=−∞ ∞ ψ_{jk} (x−l),$ for x in [0,1]. These periodized wavelets will be used for the rest of the paper

Simulation

To compare the efficiency of the block estimator under the different sampling designs, it was run on equispaced, uniform, and Poisson process samples and the results were compared. Also, the block estimator was compared to the VisuShrink estimator (VU) with universal term thresholding on uniform distributed sample points.

The test functions used are those specified in Donoho and Johnstone (1994). These eight functions represent varying degrees of spatial inhomogeneity. Sample sizes ranged from n

Function spaces

In order to define the function space F_p,q^α(M), two other spaces are needed. The first, and more general, of the two is the Besov space B_p,q^α, where 0<p,q⩽∞ and α>0. A function f is said to be in this space if its Besov norm is finite $||f||_{B_{p,q}^{α}} <∞,$ where, for 0<α⩽1 $||f||_{B_{p,q}^{α}} =||f||_{L^{p}} + ∫ 0 ∞ 1 h 1 h^{α} ||f(·+h)−f(·)||_{L^{p}}^{q} d h^{1/q}, q<∞, sup 0<h ||f(·+h)−f(·)||_{L^{p}} /h^{α}, q=∞.$ For $α>1, α=⌊α⌋+s, 0<s⩽1$ , $||f||_{B_{p,q}^{α}} = ∑ m=0 ⌊α⌋ ||f^{(m)} ||_{B_{p,q}^{s}} .$ The Hölder space Λ^α is a special case of a Besov space. Specifically, Λ^α=B_∞,∞^α. This paper

Proof

Before the main theorems are proved, some preliminaries are necessary. First, a result about the order statistics of the uniformly spaced sample points is stated.

Lemma 1

Let x_i be independent, uniform (0,1) random variables. Let x₍₁₎<x₍₂₎<⋯<x_(n) be their order statistics. Then x_(k) is a Beta (k,n−k+1) random variable with mean k/(n+1) and variance ((n+1)k−k²)/(n+1)²(n+2).

The following result from Cai (1999) will also be needed.

Lemma 2

Suppose θ_jk∈Θ_p,q^α(M) and $y_{jk} =θ_{jk} + 1 n σε_{jk}, j⩾j_{0}, k=1,2,…,2^{j},$ where z_jk are iid

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Journal of Statistical Planning and Inference

Block thresholding and wavelet estimation for nonequispaced samples

Abstract

Introduction

Section snippets

Wavelets

Simulation

Function spaces

Proof

Lower bounds on the rate of convergence of nonparametric regression estimates

J. Statist. Plann. Inference

Adaptive wavelet estimationa block thresholding and oracle inequality approach

Ann. Statist.

Wavelet shrinkage for nonequispaced samples

Ann. Statist.

Wavelet estimation for samples with random uniform design

Statist. Probab. Lett.