Besov Ball Research Articles

Wavelet estimations for densities and their derivatives with Fourier oscillating noises

By developing the classical kernel method, Delaigle and Meister provide a nice estimation for a density function with some Fourier-oscillating noises over a Sobolev ball and over risk (Delaigle and Meister in Stat. Sin. 21:1065-1092, 2011). The current paper extends their theorem to Besov ball and risk with by using wavelet methods. We firstly show a linear wavelet estimation for densities in over risk, motivated by the work of Delaigle and Meister. Our result reduces to their theorem, when . Because the linear wavelet estimator is not adaptive, a nonlinear wavelet estimator is then provided. It turns out that the convergence rate is better than the linear one for . In addition, our conclusions contain estimations for density derivatives as well.

Multichannel deconvolution with long-range dependence: A minimax study

We consider the problem of estimating the unknown response function in the multichannel deconvolution model with long-range dependent Gaussian or sub-Gaussian errors. We do not limit our consideration to a specific type of long-range dependence rather we assume that the errors should satisfy a general assumption in terms of the smallest and largest eigenvalues of their covariance matrices. We derive minimax lower bounds for the quadratic risk in the proposed multichannel deconvolution model when the response function is assumed to belong to a Besov ball and the blurring function is assumed to possess some smoothness properties, including both regular-smooth and super-smooth convolutions. Furthermore, we propose an adaptive wavelet estimator of the response function that is asymptotically optimal (in the minimax sense), or near-optimal (within a logarithmic factor), in a wide range of Besov balls, for both Gaussian and sub-Gaussian errors. It is shown that the optimal convergence rates depend on the balance between the smoothness parameter of the response function, the kernel parameters of the blurring function, the long memory parameters of the errors, and how the total number of observations is distributed among the total number of channels. Some examples of inverse problems in mathematical physics where one needs to recover initial or boundary conditions on the basis of observations from a noisy solution of a partial differential equation are used to illustrate the application of the theory we developed. The optimal convergence rates and the adaptive estimators we consider extend the ones studied by Pensky and Sapatinas (2009, 2010) for independent and identically distributed Gaussian errors to the case of long-range dependent Gaussian or sub-Gaussian errors.

Nonparametric regression estimation based on spatially inhomogeneous data: minimax global convergence rates and adaptivity

We consider the nonparametric regression estimation problem of recovering an unknown response function f on the basis of spatially inhomogeneous data when the design points follow a known density g with a finite number of well-separated zeros. In particular, we consider two different cases: when g has zeros of a polynomial order and when g has zeros of an exponential order. These two cases correspond to moderate and severe data losses, respectively. We obtain asymptotic (as the sample size increases) minimax lower bounds for the L 2 -risk when f is assumed to belong to a Besov ball, and construct adaptive wavelet thresholding estimators of f that are asymptotically optimal (in the minimax sense) or near-optimal within a logarithmic factor (in the case of a zero of a polynomial order), over a wide range of Besov balls. The spatially inhomogeneous ill-posed problem that we investigate is inherently more difficult than spatially homogeneous ill-posed problems like, e.g., deconvolution. In particular, due to spatial irregularity, assessment of asymptotic minimax global convergence rates is a much harder task than the derivation of asymptotic minimax local convergence rates studied recently in the literature. Furthermore, the resulting estimators exhibit very different behavior and asymptotic minimax global convergence rates in comparison with the solution of spatially homogeneous ill-posed problems. For example, unlike in the deconvolution problem, the asymptotic minimax global convergence rates are greatly influenced not only by the extent of data loss but also by the degree of spatial homogeneity of f .S pecif ically, even if 1/g is non-integrable, one can recover f as well as in the case of an equispaced design (in terms of asymptotic minimax global convergence rates) when it is homogeneous enough since the estimator is borrowing strength in the areas where f is adequately sampled.

Wavelet optimal estimations for a density with some additive noises

Using wavelet methods, Fan and Koo study optimal estimations for a density with some additive noises over a Besov ball Br,qs(L) (r,q⩾1) and over L2 risk (Fan and Koo, 2002 [13]). The L∞ risk estimations are investigated by Lounici and Nickl (2011) [19]. This paper deals with optimal estimations over Lp (1⩽p⩽∞) risk for moderately ill-posed noises. A lower bound of Lp risk is firstly provided, which generalizes Fan–Koo and Lounici–Nickl's theorems; then we define a linear and non-linear wavelet estimators, motivated by Fan–Koo and Pensky–Vidakovic's work. The linear one is rate optimal for r⩾p, and the non-linear estimator attains suboptimal (optimal up to a logarithmic factor). These results can be considered as an extension of some theorems of Donoho et al. (1996) [10]. In addition, our non-linear wavelet estimator is adaptive to the indices s, r, q and L.

Anisotropic de-noising in functional deconvolution model with dimension-free convergence rates

In the present paper we consider the problem of estimating a periodic $(r+1)$-dimensional function $f$ based on observations from its noisy convolution. We construct a wavelet estimator of $f$, derive minimax lower bounds for the $L^{2}$-risk when $f$ belongs to a Besov ball of mixed smoothness and demonstrate that the wavelet estimator is adaptive and asymptotically near-optimal within a logarithmic factor, in a wide range of Besov balls. We prove in particular that choosing this type of mixed smoothness leads to rates of convergence which are free of the “curse of dimensionality” and, hence, are higher than usual convergence rates when $r$ is large. The problem studied in the paper is motivated by seismic inversion which can be reduced to solution of noisy two-dimensional convolution equations that allow to draw inference on underground layer structures along the chosen profiles. The common practice in seismology is to recover layer structures separately for each profile and then to combine the derived estimates into a two-dimensional function. By studying the two-dimensional version of the model, we demonstrate that this strategy usually leads to estimators which are less accurate than the ones obtained as two-dimensional functional deconvolutions. Indeed, we show that unless the function $f$ is very smooth in the direction of the profiles, very spatially inhomogeneous along the other direction and the number of profiles is very limited, the functional deconvolution solution has a much better precision compared to a combination of $M$ solutions of separate convolution equations. A limited simulation study in the case of $r=1$ confirms theoretical claims of the paper.

Minimax convergence rates under the [formula omitted]-risk in the functional deconvolution model

We derive minimax results in the functional deconvolution model under the L p -risk, 1 ≤ p < ∞ . Lower bounds are given when the unknown response function is assumed to belong to a Besov ball and under appropriate smoothness assumptions on the blurring function, including both regular-smooth and super-smooth convolutions. Furthermore, we investigate the asymptotic minimax properties of an adaptive wavelet estimator over a wide range of Besov balls. The new findings extend recently obtained results under the L 2 -risk. As an illustration, we discuss particular examples for both continuous and discrete settings.

Functional deconvolution in a periodic setting: Uniform case

We extend deconvolution in a periodic setting to deal with functional data. The resulting functional deconvolution model can be viewed as a generalization of a multitude of inverse problems in mathematical physics where one needs to recover initial or boundary conditions on the basis of observations from a noisy solution of a partial differential equation. In the case when it is observed at a finite number of distinct points, the proposed functional deconvolution model can also be viewed as a multichannel deconvolution model. We derive minimax lower bounds for the $L^2$-risk in the proposed functional deconvolution model when $f(\cdot)$ is assumed to belong to a Besov ball and the blurring function is assumed to possess some smoothness properties, including both regular-smooth and super-smooth convolutions. Furthermore, we propose an adaptive wavelet estimator of $f(\cdot)$ that is asymptotically optimal (in the minimax sense), or near-optimal within a logarithmic factor, in a wide range of Besov balls. In addition, we consider a discretization of the proposed functional deconvolution model and investigate when the availability of continuous data gives advantages over observations at the asymptotically large number of points. As an illustration, we discuss particular examples for both continuous and discrete settings.

A complement to Le Cam’s theorem

This paper examines asymptotic equivalence in the sense of Le Cam between density estimation experiments and the accompanying Poisson experiments. The significance of asymptotic equivalence is that all asymptotically optimal statistical procedures can be carried over from one experiment to the other. The equivalence given here is established under a weak assumption on the parameter space $\mathcal{F}$. In particular, a sharp Besov smoothness condition is given on $\mathcal{F}$ which is sufficient for Poissonization, namely, if $\mathcal{F}$ is in a Besov ball $B_{p,q}^{\alpha}(M)$ with $\alpha p>1/2$. Examples show Poissonization is not possible whenever $\alpha p 0$.

Nonlinear estimation over weak Besov spaces and minimax Bayes method

Weak Besov spaces play important roles in statistics as maxisets of classical procedures or for measuring the sparsity of signals. The goal of this paper is to study weak Besov balls ${\cal WB}_{s,p,q}(C)$ from the statistical point of view by using the minimax Bayes method. In particular, we compare weak and strong Besov balls statistically. By building an optimal Bayes wavelet thresholding rule, we first establish that, under suitable conditions, the rate of convergence of the minimax risk for ${\cal WB}_{s,p,q}(C)$ is the same as for the strong Besov ball ${\cal B}_{s,p,q}(C)$ that is contained by ${\cal WB}_{s,p,q}(C)$. However, we show that the asymptotically least favorable priors of ${\cal WB}_{s,p,q}(C)$ that are based on Pareto distributions cannot be asymptotically least favorable priors for ${\cal B}_{s,p,q}(C)$. Finally, we present sample paths of such priors that provide representations of the worst functions to be estimated for classical procedures and we give an interpretation of the roles of the parameters $s$, $p$ and $q$ of ${\cal WB}_{s,p,q}(C)$.

Pointwise optimality of Bayesian wavelet estimators

We consider pointwise mean squared errors of several known Bayesian wavelet estimators, namely, posterior mean, posterior median and Bayes Factor, where the prior imposed on wavelet coefficients is a mixture of an atom of probability zero and a Gaussian density. We show that for the properly chosen hyperparameters of the prior, all the three estimators are (up to a log-factor) asymptotically minimax within any prescribed Besov ball $$B^{s}_{p},_{q} (M)$$ . We discuss the Bayesian paradox and compare the results for the pointwise squared risk with those for the global mean squared error.

Multiple Wavelet Basis Image Denoising Using Besov Ball Projections

We propose a new image denoising algorithm that exploits an image's representation in multiple wavelet domains. Besov balls are convex sets of images whose Besov norms are bounded from above by their radii. Projecting an image onto a Besov ball of proper radius corresponds to a type of wavelet shrinkage for image denoising. By defining Besov balls in multiple wavelet domains and projecting onto their intersection using the projection onto convex sets (POCS) algorithm, we obtain an estimate that effectively combines estimates from multiple wavelet domains. While simple, the algorithm provides significant improvement over conventional wavelet shrinkage algorithms based on a single wavelet domain.