Wavelet Shrinkage with Double Weibull Prior

In this article, we propose a denoising methodology in the wavelet domain based on a Bayesian hierarchical model using Double Weibull prior. We propose two estimators, one based on posterior mean (Double Weibull Wavelet Shrinker, DWWS) and the other based on larger posterior mode (DWWS-LPM), and show how to calculate them efficiently. Traditionally, mixture priors have been used for modeling sparse wavelet coefficients. The interesting feature of this article is the use of non-mixture prior. We show that the methodology provides good denoising performance, comparable even to state-of-the-art methods that use mixture priors and empirical Bayes setting of hyperparameters, which is demonstrated by extensive simulations on standardly used test functions. An application to real-word dataset is also considered.     

Wavelet density estimation for stratified size-biased sample

In this paper we consider the estimation of a density function on the basis of a random stratified sample from weighted distributions. We propose a linear wavelet density estimator and prove its consistency. The behavior of the proposed estimator and its smoothed versions is eventually illustrated by simulated examples and a case study involving alcohol blood level in DUI cases.     

Wavelet‐based estimators for mixture regression

We 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.     

Bayesian False Discovery Rate Wavelet Shrinkage: Theory and Applications

Statistical inference in the wavelet domain remains a vibrant area of contemporary statistical research because of desirable properties of wavelet representations and the need of scientific community to process, explore, and summarize massive data sets. Prime examples are biomedical, geophysical, and internet related data. We propose two new approaches to wavelet shrinkage/thresholding.

In the spirit of Efron and Tibshirani's recent work on local false discovery rate, we propose Bayesian Local False Discovery Rate (BLFDR), where the underlying model on wavelet coefficients does not assume known variances. This approach to wavelet shrinkage is shown to be connected with shrinkage based on Bayes factors. The second proposal, Bayesian False Discovery Rate (BaFDR), is based on ordering of posterior probabilities of hypotheses on true wavelets coefficients being null, in Bayesian testing of multiple hypotheses.

We demonstrate that both approaches result in competitive shrinkage methods by contrasting them to some popular shrinkage techniques.     

Variance Estimation for Fractional Brownian Motions with Fixed Hurst Parameters

Some real-world phenomena in geo-science, micro-economy, and turbulence, to name a few, can be effectively modeled by a fractional Brownian motion indexed by a Hurst parameter, a regularity level, and a scaling parameter σ², an energy level. This article discusses estimation of a scaling parameter σ² when a Hurst parameter is known. To estimate σ², we propose three approaches based on maximum likelihood estimation, moment-matching, and concentration inequalities, respectively, and discuss the theoretical characteristics of the estimators and optimal-filtering guidelines. We also justify the improvement of the estimation of σ² when a Hurst parameter is known. Using the three approaches and a parametric bootstrap methodology in a simulation study, we compare the confidence intervals of σ² in terms of their lengths, coverage rates, and computational complexity and discuss empirical attributes of the tested approaches. We found that the approach based on maximum likelihood estimation was optimal in terms of efficiency and accuracy, but computationally expensive. The moment-matching approach was found to be not only comparably efficient and accurate but also computationally fast and robust to deviations from the fractional Brownian motion model.     

Lower bounds on Bayes risks for estimating a normal variance: With applications

Brown and Gajek (1990) gave useful lower bounds on Bayes risks, which improve on earlier bounds by various authors. Many of these use the information inequality. For estimating a normal variance using the invariant quadratic loss and any arbitrary prior on the reciprocal of the variance that is a mixture of Gamma distributions, we obtain lower bounds on Bayes risks that are different from Borovkov-Sakhanienko bounds. The main tool is convexity of appropriate functionals as opposed to the information inequality. The bounds are then applied to many specific examples, including the multi-Bayesian setup (Zidek and his coauthors). Subsequent use of moment theory and geometry gives a number of new results on efficiency of estimates which are linear in the sufficient statistic. These results complement earlier results of Donoho, Liu and MacGibbon (1990), Johnstone and MacGibbon (1992) and Vidakovic and DasGupta (1994) for the location case.     

Larger posterior mode wavelet thresholding and applications

This paper explores the thresholding rules induced by a variation of the Bayesian MAP principle. The MAP rules are Bayes actions that maximize the posterior. The proposed rule is thresholding and always picks the mode of the posterior larger in absolute value, thus the name LPM. We demonstrate that the introduced shrinkage performs comparably to several popular shrinkage techniques. The exact risk properties of the thresholding rule are explored, as well. We provide extensive simulational analysis and apply the proposed methodology to real-life experimental data coming from the field of atomic force microscopy.