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

This paper studies the pointwise estimation of copula density c under a censoring model. We use the wavelet method to construct a corresponding linear wavelet estimator with the new weight, and provide the risk bound for the pointwise error of the estimator on the Besov space B ∞ s . The results show that our convergence rate is optimal. Finally, the numerical results evaluate the feasibility and effectiveness of our proposed estimator.

Pointwise density estimation for biased 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.

Seemingly unrelated regression models (SURMs) are extensions of linear regression models which allow correlated errors between regression equations. The purpose of this article is to reconsider some fundamental problems on the performance and connection of ordinary least-squares estimators (OLSEs) and the best linear unbiased estimators (BLUEs) of parametric functions under an SURM. Motivated by a variety of known results and facts on the equivalence of OLSEs and BLUEs under general linear models, this article collects a list of necessary and sufficient conditions for OLSEs to be BLUEs under an SURM and presents a variety of statistical interpretations on the equivalence of OLSEs and BLUEs under the SURM.

Using compactly supported wavelets, Giné and Nickl [Uniform limit theorems for wavelet density estimators, Ann. Probab. 37(4) (2009) 1605–1646] obtain the optimal strong [Formula: see text] convergence rates of wavelet estimators for a fixed noise-free density function. They also study the same problem by spline wavelets [Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections, Bernoulli 16(4) (2010) 1137–1163]. This paper considers the strong [Formula: see text] convergence of wavelet deconvolution density estimators. We first show the strong [Formula: see text] consistency of our wavelet estimator, when the Fourier transform of the noise density has no zeros. Then strong [Formula: see text] convergence rates are provided, when the noises are severely and moderately ill-posed. In particular, for moderately ill-posed noises and [Formula: see text], our convergence rate is close to Giné and Nickl’s.

Many efficient wavelet domain estimation algorithms for noise reduction have been proposed in the literature. All of them are based on variance estimation, which require all wavelet coefficients to be saved in the memory thus causing constrains towards the development of real time system. Tarp filter has been used successfully for online/real-time variance estimation of the multilevel wavelets and has shown exceptional performance in image compression. This paper proposes the utilization of the Tarp filter for the estimation of wavelet coefficient variance without saving them in memory. The paper presents the results for Tarp-linear estimator (TLE) and compares it with simple linear estimator (LE). It is found that TLE, while having an advantage of memory free noise reduction method, compromise on SNR

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.

Adaptive nonparametric empirical Bayes estimation via wavelet series: The minimax study

On the equivalence of the BLUEs under a general linear model and its restricted and stochastically restricted models

Longitudinal data, in which repeated responses of each individual are collected at different times or under different conditions, are not uncommon in many fields. It is common to assume that the responses of each individual are normally distributed. This assumption may be violated in practice for many reasons. These include outliers, contaminated data, and heavy-tailed distributions. Hence, there is a need for techniques that capable of dealing with such non-normal responses. The adaptive linear regression estimator (ALR) proposed mainly to handle non-normal cross-sectional data. It is a linear combination of ordinary least squares method (OLS) and least absolute deviations method (LAD). In this article the adaptive linear regression estimator (ALR) is modified and developed to deal with heavy-tailed longitudinal data. The asymptotic properties of modified adaptive linear regression estimator (MALR) are also derived. The MALR estimator is efficient for both light-tailed and heavy-tailed distributions. Simulation studies are conducted to evaluate the modified approach. Also, the new approach is applied to real data that represent measurements of the amount of pain experienced during labor.

ABSTRACTThe estimation of a multivariate function from a stationary m-dependent process is investigated, with a special focus on the case where m is large or unbounded. We develop an adaptive estimator based on wavelet methods. Under flexible assumptions on the nonparametric model, we prove the good performances of our estimator by determining sharp rates of convergence under two kinds of errors: the pointwise mean squared error and the mean integrated squared error. We illustrate our theoretical result by considering the multivariate density estimation problem, the derivatives density estimation problem, the density estimation problem in a GARCH-type model and the multivariate regression function estimation problem. The performance of proposed estimator has been shown by a numerical study for a simulated and real data sets.

Nonparametric hazard rate estimation by orthogonal wavelet methods

In this paper we investigate the robust estimation of generalized varying coefficient models in which the unknown regression coefficients may change with different explanatory variables. Based on the B-spline series approximation and Walsh-average technique we develop an initial estimator for the unknown regression coefficient functions. By virtue of the initial estimator, the generalized varying coefficient model is reduced to a univariate nonparametric regression model. Then combining the local linear smooth and Walsh average technique we further propose a two-stage local linear Walsh-average estimator for the unknown regression coefficient functions. Under mild assumptions, we establish the large sample theory of the proposed estimators by utilizing the results of U-statistics and shows that the two-stage local linear Walsh-average estimator own an oracle property, namely the asymptotic normality of the two-stage local linear Walsh-average estimator of each coefficient function is not affected by other unknown coefficient functions. Extensive simulation studies are conducted to assess the finite sample performance, and a real example is analyzed to illustrate the proposed method.

A multi-rate hierarchical fault-tolerant adaptive model predictive control framework: Theory and design for quadrotors

Minimax estimation of the diffusion coefficient through irregular samplings