User-Friendly Covariance Estimation for Heavy-Tailed Distributions

We offer a survey of recent results on covariance estimation for heavy-tailed distributions. By unifying ideas scattered in the literature, we propose user-friendly methods that facilitate practical implementation. Specifically, we introduce element-wise and spectrum-wise truncation operators, as well as their $M$-estimator counterparts, to robustify the sample covariance matrix. Different from the classical notion of robustness that is characterized by the breakdown property, we focus on the tail robustness which is evidenced by the connection between nonasymptotic deviation and confidence level. The key observation is that the estimators needs to adapt to the sample size, dimensionality of the data and the noise level to achieve optimal tradeoff between bias and robustness. Furthermore, to facilitate their practical use, we propose data-driven procedures that automatically calibrate the tuning parameters. We demonstrate their applications to a series of structured models in high dimensions, including the bandable and low-rank covariance matrices and sparse precision matrices. Numerical studies lend strong support to the proposed methods.Comment: 56 pages, 2 figure

Morphological Dependence of Star Formation Properties for the Galaxies in the Merging Galaxy Cluster A2255

The merging cluster of galaxies A2255 is covered by the Sloan Digital Sky Survey (SDSS) survey. In this paper we perform a morphological classification on the basis of the SDSS imaging and spectral data, and investigate the morphological dependence of the star formation rates (SFRs) for these member galaxies. As we expect, a tight correlation between the normalized SFR by stellar mass (SFR/M$_*$) and the H$\alpha$ equivalent width is found for the late-type galaxies in A2255. The correlation of SFR/M$_*$ with the continuum break strength at 4000 \AA is also confirmed. The SFR/M$_*$ - M$_*$ correlation is found for both the early- and late-type galaxies, indicating that the star formation activity tends to be suppressed when the assembled stellar mass M$_*$) increases, and this correlation is tighter and steeper for the late-type cluster galaxies. Compared with the mass range of field spiral galaxies, only two massive late-type galaxies with M$_*>10^{11}$ M$_{\odot}$ are survived in A2255, suggesting that the gas disks of massive spiral galaxies could have been tidally stripped during cluster formation. Additionally, the SFR variation with the projected radial distance are found to be heavily dependent upon galaxy morphology: the early-type galaxies have a very weak inner decrease in SFR/M$_*$, while the inner late-type galaxies tend to have higher SFR/M$_*$ values than the outer late-types. This may suggest that the galaxy-scale turbulence stimulated by the merging of subclusters might have played different roles on early- and late-type galaxies, which leads to a suppression of the star formation activity for E/S0 galaxies and a SFR enhancement for spiral and irregular galaxies.Comment: 21 pages, including 7 EPS figures and 1 tables, uses aastex.cls, Accepted by the A

BEAUTY Powered BEAST

We study inference about the uniform distribution with the proposed binary expansion approximation of uniformity (BEAUTY) approach. Through an extension of the celebrated Euler's formula, we approximate the characteristic function of any copula distribution with a linear combination of means of binary interactions from marginal binary expansions. This novel characterization enables a unification of many important existing tests through an approximation from some quadratic form of symmetry statistics, where the deterministic weight matrix characterizes the power properties of each test. To achieve a uniformly high power, we study test statistics with data-adaptive weights through an oracle approach, referred to as the binary expansion adaptive symmetry test (BEAST). By utilizing the properties of the binary expansion filtration, we show that the Neyman-Pearson test of uniformity can be approximated by an oracle weighted sum of symmetry statistics. The BEAST with this oracle leads all existing tests we considered in empirical power against all complex forms of alternatives. This oracle therefore sheds light on the potential of substantial improvements in power and on the form of optimal weights under each alternative. By approximating this oracle with data-adaptive weights, we develop the BEAST that improves the empirical power of many existing tests against a wide spectrum of common alternatives while providing clear interpretation of the form of non-uniformity upon rejection. We illustrate the BEAST with a study of the relationship between the location and brightness of stars