Posterior mean and variance approximation for regression and time series problems

This paper develops a methodology for approximating the posterior first two moments of the posterior distribution in Bayesian inference. Partially specified probability models that are defined only by specifying means and variances, are constructed based upon second-order conditional independence in order to facilitate posterior updating and prediction of required distributional quantities. Such models are formulated particularly for multivariate regression and time series analysis with unknown observational variance-covariance components. The similarities and differences of these models with the Bayes linear approach are established. Several subclasses of important models, including regression and time series models with errors following multivariate t, inverted multivariate t and Wishart distributions, are discussed in detail. Two numerical examples consisting of simulated data and of US investment and change in inventory data illustrate the proposed methodology

Estimation of Single-Index Models Based on Boosting Techniques

In single-index models the link or response function is not considered as fixed. The data determine the form of the unknown link function. In order to obtain a flexible form of the link function we specify the link function as an expansion in basis function and propose to estimate parameters as well as the link function by weak learners within a boosting framework. It is shown that the method is a strong competitor to existing methods. The method is investigated in simulation studies and applied to real data

Off-the-Grid Line Spectrum Denoising and Estimation with Multiple Measurement Vectors

Compressed Sensing suggests that the required number of samples for reconstructing a signal can be greatly reduced if it is sparse in a known discrete basis, yet many real-world signals are sparse in a continuous dictionary. One example is the spectrally-sparse signal, which is composed of a small number of spectral atoms with arbitrary frequencies on the unit interval. In this paper we study the problem of line spectrum denoising and estimation with an ensemble of spectrally-sparse signals composed of the same set of continuous-valued frequencies from their partial and noisy observations. Two approaches are developed based on atomic norm minimization and structured covariance estimation, both of which can be solved efficiently via semidefinite programming. The first approach aims to estimate and denoise the set of signals from their partial and noisy observations via atomic norm minimization, and recover the frequencies via examining the dual polynomial of the convex program. We characterize the optimality condition of the proposed algorithm and derive the expected convergence rate for denoising, demonstrating the benefit of including multiple measurement vectors. The second approach aims to recover the population covariance matrix from the partially observed sample covariance matrix by motivating its low-rank Toeplitz structure without recovering the signal ensemble. Performance guarantee is derived with a finite number of measurement vectors. The frequencies can be recovered via conventional spectrum estimation methods such as MUSIC from the estimated covariance matrix. Finally, numerical examples are provided to validate the favorable performance of the proposed algorithms, with comparisons against several existing approaches.Comment: 14 pages, 10 figure

Sparse component separation for accurate CMB map estimation

The Cosmological Microwave Background (CMB) is of premier importance for the cosmologists to study the birth of our universe. Unfortunately, most CMB experiments such as COBE, WMAP or Planck do not provide a direct measure of the cosmological signal; CMB is mixed up with galactic foregrounds and point sources. For the sake of scientific exploitation, measuring the CMB requires extracting several different astrophysical components (CMB, Sunyaev-Zel'dovich clusters, galactic dust) form multi-wavelength observations. Mathematically speaking, the problem of disentangling the CMB map from the galactic foregrounds amounts to a component or source separation problem. In the field of CMB studies, a very large range of source separation methods have been applied which all differ from each other in the way they model the data and the criteria they rely on to separate components. Two main difficulties are i) the instrument's beam varies across frequencies and ii) the emission laws of most astrophysical components vary across pixels. This paper aims at introducing a very accurate modeling of CMB data, based on sparsity, accounting for beams variability across frequencies as well as spatial variations of the components' spectral characteristics. Based on this new sparse modeling of the data, a sparsity-based component separation method coined Local-Generalized Morphological Component Analysis (L-GMCA) is described. Extensive numerical experiments have been carried out with simulated Planck data. These experiments show the high efficiency of the proposed component separation methods to estimate a clean CMB map with a very low foreground contamination, which makes L-GMCA of prime interest for CMB studies.Comment: submitted to A&

Treatment Effects on Ordinal Outcomes: Causal Estimands and Sharp Bounds

Assessing the causal effects of interventions on ordinal outcomes is an important objective of many educational and behavioral studies. Under the potential outcomes framework, we can define causal effects as comparisons between the potential outcomes under treatment and control. However, unfortunately, the average causal effect, often the parameter of interest, is difficult to interpret for ordinal outcomes. To address this challenge, we propose to use two causal parameters, which are defined as the probabilities that the treatment is beneficial and strictly beneficial for the experimental units. However, although well-defined for any outcomes and of particular interest for ordinal outcomes, the two aforementioned parameters depend on the association between the potential outcomes, and are therefore not identifiable from the observed data without additional assumptions. Echoing recent advances in the econometrics and biostatistics literature, we present the sharp bounds of the aforementioned causal parameters for ordinal outcomes, under fixed marginal distributions of the potential outcomes. Because the causal estimands and their corresponding sharp bounds are based on the potential outcomes themselves, the proposed framework can be flexibly incorporated into any chosen models of the potential outcomes, and are directly applicable to randomized experiments, unconfounded observational studies, and randomized experiments with noncompliance. We illustrate our methodology via numerical examples and three real-life applications related to educational and behavioral research.Comment: Accepted by the Journal of Education and Behavioral Statistic