Linear Wavelet Estimator Research Articles

Wavelet estimations of a density function in two-class mixture model

This paper considers nonparametric estimations of a density function in a two-class mixture model. A linear wavelet estimator and an adaptive wavelet estimator are constructed. Upper bound estimations over $ L^{p}\; (1\leq p < +\infty) $ risk of those wavelet estimators are proved in Besov spaces. When $ \tilde{p}\geq p\geq1 $, the convergence rate of adaptive wavelet estimator is the same as the linear estimator up to a $ \ln n $ factor. The adaptive wavelet estimator can get better than the linear estimator in the case of $ 1\leq \tilde{p} < p $. Finally, some numerical experiments are presented to validate the theoretical results.

Pointwise Estimation of Anisotropic Regression Functions Using Wavelets with Data-Driven Selection Rule

For nonparametric regression estimation, conventional research all focus on isotropic regression function. In this paper, a linear wavelet estimator of anisotropic regression function is constructed, the rate of convergence of this estimator is discussed in anisotropic Besov spaces. More importantly, in order to obtain an adaptive estimator, a regression estimator is proposed with scaling parameter data-driven selection rule. It turns out that our results attain the optimal convergence rate of nonparametric pointwise estimation.

Wavelet Estimation of Function Derivatives from a Multichannel Deconvolution Model

This paper considers a multichannel deconvolution model with Gaussian white noises. The goal is to estimate the d -th derivatives of an unknown function in the model. For super-smooth case, we construct an adaptive linear wavelet estimator by wavelet projection method. For regular-smooth case, we provide an adaptive nonlinear wavelet estimator by hard-thresholded method. In order to measure the global performances of our estimators, we show upper bounds on convergence rates using the L p -risk ( 1 ≤ p < ∞ ).

Pointwise Wavelet Estimations for a Regression Model in Local Hölder Space

This paper considers an unknown functional estimation problem in a regression model with multiplicative and additive noise. A linear wavelet estimator is first constructed by a wavelet projection operator. The convergence rate under the pointwise error of linear wavelet estimators is studied in local Hölder space. A nonlinear wavelet estimator is provided by the hard thresholding method in order to obtain an adaptive estimator. The convergence rate of the nonlinear estimator is the same as the linear estimator up to a logarithmic term. Finally, it should be pointed out that the convergence rates of two wavelet estimators are consistent with the optimal convergence rate on pointwise nonparametric estimation.

Wavelet-based estimation in a semiparametric regression model

In this paper, we introduce a wavelet-based method for estimating the effective dimension reduction (EDR) space in the semiparametric regression model introduced by Li [Sliced inverse regression for dimension reduction, J. Amer. Statist. Assoc. 86 (1991) 316–327]. This method is obtained by using linear wavelet estimators of the density and regression functions that are involved in the covariance matrix of conditional expectation whose eigenvectors span the EDR space. Then, consistency of the proposed estimators is proved. A simulation study that allows one to evaluate the performance of the proposal with comparison to existing methods is presented.

Spectral-free estimation of Lévy densities in high-frequency regime

We construct an estimator of the Lévy density of a pure jump Lévy process, possibly of infinite variation, from the discrete observation of one trajectory at high frequency. The novelty of our procedure is that we directly estimate the Lévy density relying on a pathwise strategy, whereas existing procedures rely on spectral techniques. By taking advantage of a compound Poisson approximation, we circumvent the use of spectral techniques and in particular of the Lévy–Khintchine formula. A linear wavelet estimator is built and its performance is studied in terms of Lp loss functions, p≥1, over Besov balls. We recover classical nonparametric rates for finite variation Lévy processes and for a nonparametric class of symmetric infinite variation Lévy processes. We show that the procedure is robust when the estimation set gets close to the critical value 0 and also discuss its robustness to the presence of a Brownian part.

The optimal wavelet estimation for biased density

In this paper, the optimal estimation at a given point is considered for biased density in theory. A lower bound is first provided for all possible estimators over the local Hölder space. Using wavelet method, a linear wavelet estimator is constructed and turned out to be optimal by investigating its upper bound of pointwise convergence rate. To get the adaptivity, a nonlinear wavelet estimator is obtained and proved to be near-optimal within a logarithmic factor.

A Note on the Nonparametric Estimation of the Conditional Mode by Wavelet Methods

The purpose of this note is to introduce and investigate the nonparametric estimation of the conditional mode using wavelet methods. We propose a new linear wavelet estimator for this problem. The estimator is constructed by combining a specific ratio technique and an established wavelet estimation method. We obtain rates of almost sure convergence over compact subsets of Rd. A general estimator beyond the wavelet methodology is also proposed, discussing adaptivity within this statistical framework.

Point-Wise Wavelet Estimation in the Convolution Structure Density Model

By using a kernel method, Lepski and Willer establish adaptive and optimal $$L^p$$ risk estimations in the convolution structure density model in 2017 and 2019. They assume their density functions to be in a Nikol’skii space. Motivated by their work, we first use a linear wavelet estimator to obtain a point-wise optimal estimation in the same model. We allow our densities to be in a local and anisotropic Holder space. Then a data driven method is used to obtain an adaptive and near-optimal estimation. Finally, we show the logarithmic factor necessary to get the adaptivity.

Convergence Rates of Wavelet Density Estimators for Strongly Mixing Samples

This paper considers wavelet estimations for a multivariate density function based on strongly mixing data. We first construct a linear wavelet estimator and provide a convergence rate over $$L^{p} (1\le p<\infty )$$ risk in Besov space $$B^{s}_{r,q}(\mathbb {R}^{d})$$ . However, this estimator depends on the smoothness of density function, which means that the estimator is not adaptive. A nonlinear adaptive wavelet estimator is proposed by thresholding method. Moreover, the convergence rate of nonlinear estimator is better than the linear one in the case of $$r\le p$$ .

Wavelet regression estimation over Lp risk based on negatively associated sample

This paper considers wavelet estimations of a regression function based on negatively associated sample. We provide upper bound estimations over [Formula: see text] risk of linear and nonlinear wavelet estimators in Besov space, respectively. When the random sample reduces to the independent case, our convergence rates coincide with the optimal convergence rates of classical nonparametric regression estimation.

Wavelet pointwise estimations under multiplicative censoring

In this paper, we investigate the wavelet-based estimators of a kind of censored mixture density and discuss their pointwise asymptotic convergence rates over Hölder spaces. We first consider the linear wavelet estimator and give its upper bound. However, the linear one is nonadaptive and not applicable since it is related to the unknown space parameter. Finally, we use the hard threshold method to explore adaptive nonlinear wavelet estimator and obtain the same convergence order as the linear one up to a logarithmic penalty.

MISE of wavelet estimators for regression derivatives with biased strong mixing data

Using a wavelet basis, this article considers the mean integrated squared error (MISE) of linear and non linear wavelet estimators for regression derivatives based on biased strong mixing data. It turns out that the convergence rates coincide with those of Chesneau and Shirazi’s (Communication in Statistics-Theory and Methods, 2014), when d = 0 and the random sample is independent.

Pointwise density estimation based on negatively associated data

In this paper, we consider pointwise estimation over l^{p}(1leq p<infty ) risk for a density function based on a negatively associated sample. We construct linear and nonlinear wavelet estimators and provide their convergence rates. It turns out that those wavelet estimators have the same convergence rate up to the lnn factor. Moreover, the nonlinear wavelet estimator is adaptive.

Nonparametric Estimation for Hazard Rate Function by Wavelet Procedures with Simulation

This article is a study of a non-parametric estimation of the hazard rate function using the linear wavelet estimation for right randomly censoring data. The strategy of the estimation is based on the use of the wavelet projection of the father function{φj,k(x),j = 1,2,…, 0 ≤ k ≤2j−1} on the subspace (Vj) of the spaceL2(R), with the Breslow estimate of the cumulative function. Real data of patients suffering from liver metastases is using as an application. Moreover, a simulation study is used to give more clarify to the method of estimation.

Pointwise Wavelet Estimation of Regression Function Based on Biased Data

In this paper, we establish wavelet estimations on pointwise $$l^{p}(1\le p<\infty )$$ risk for multivariate regression functions based on biased data. We firstly introduce a linear wavelet estimator and discuss the convergence rate of this estimator. In order to obtain an adaptive estimator, a nonlinear wavelet estimator is constructed by the hard thresholding method. It should be pointed out that the convergence rate of linear and nonlinear wavelet estimators coincide with the optimal convergence rate of pointwise nonparametric estimation.

Convergence rates of wavelet density estimation for negatively dependent sample

In this paper, linear and nonlinear wavelet estimators are defined for a density in a Besov space based on a negatively dependent random sample, and their upper bounds on L^{p} (1leq p<infty ) risk are provided.

Non linear wavelet estimation of regression derivatives based on biased data

This paper considers non linear wavelet estimation on risk for the derivatives of regression function in Besov spaces based on biased data. It turns out that the convergence rate of non linear wavelet estimator keeps the same as Chaubey, Chesneau and Navarro’s linear estimator (Linear wavelet estimation of the derivatives of a regression function based on biased data, Communication in Statistics-Theory and Methods, 46: 9541-9556, 2017), when . However, the non linear estimator gets better if . On the other hand, our estimator is adaptive. Finally, our theory is illustrated with a simulation study.

Wavelet optimal estimations for a two-dimensional continuous-discrete density function over L^{p} risk

The mixed continuous-discrete density model plays an important role in reliability, finance, biostatistics, and economics. Using wavelets methods, Chesneau, Dewan, and Doosti provide upper bounds of wavelet estimations on L^{2} risk for a two-dimensional continuous-discrete density function over Besov spaces B^{s}_{r,q}. This paper deals with L^{p} (1leq p < infty) risk estimations over Besov space, which generalizes Chesneau–Dewan–Doosti’s theorems. In addition, we firstly provide a lower bound of L^{p} risk. It turns out that the linear wavelet estimator attains the optimal convergence rate for r geq p, and the nonlinear one offers optimal estimation up to a logarithmic factor.

Wavelet‐based estimators for mixture regression

AbstractWe consider a process that is observed as a mixture of two random distributions, where the mixing probability is an unknown function of time. The setup is built upon a wavelet‐based mixture regression. Two linear wavelet estimators are proposed. Furthermore, we consider three regularizing procedures for each of the two wavelet methods. We also discuss regularity conditions under which the consistency of the wavelet methods is attained and derive rates of convergence for the proposed estimators. A Monte Carlo simulation study is conducted to illustrate the performance of the estimators. Various scenarios for the mixing probability function are used in the simulations, in addition to a range of sample sizes and resolution levels. We apply the proposed methods to a data set consisting of array Comparative Genomic Hybridization from glioblastoma cancer studies.