Matrix measures on the unit circle, moment spaces, orthogonal polynomials and the Geronimus relations

We study the moment space corresponding to matrix measures on the unit circle. Moment points are characterized by non-negative definiteness of block Toeplitz matrices. This characterization is used to derive an explicit representation of orthogonal polynomials with respect to matrix measures on the unit circle and to present a geometric definition of canonical moments. It is demonstrated that these geometrically defined quantities coincide with the Verblunsky coefficients, which appear in the Szeg\"{o} recursions for the matrix orthogonal polynomials. Finally, we provide an alternative proof of the Geronimus relations which is based on a simple relation between canonical moments of matrix measures on the interval [-1,1] and the Verblunsky coefficients corresponding to matrix measures on the unit circle.Comment: 25 page

Bridge estimators and the adaptive Lasso under heteroscedasticity

In this paper we investigate penalized least squares methods in linear regression models with heteroscedastic error structure. It is demonstrated that the basic properties with respect to model selection and parameter estimation of bridge estimators, Lasso and adaptive Lasso do not change if the assumption of homoscedasticity is violated. However, these estimators do not have oracle properties in the sense of Fan and Li (2001). In order to address this problem we introduce weighted penalized least squares methods and demonstrate their advantages by asymptotic theory and by means of a simulation study

The adaptive Lasso in high dimensional sparse heteroscedastic models

In this paper we study the asymptotic properties of the adaptive Lasso estimate in high dimensional sparse linear regression models with heteroscedastic errors. It is demonstrated that model selection properties and asymptotic normality of the selected parameters remain valid but with a suboptimal asymptotic variance. A weighted adaptive Lasso estimate is introduced and is investigated. In particular, it is shown that the new estimate performs consistent model selection and that linear combinations of the estimates corresponding to the non-vanishing components are asymptotically normally distributed with a smaller variance than those obtained by the "classical" adaptive Lasso. The results are illustrated in a data example and by means of a small simulation study

Testing for a constant coefficient of variation in nonparametric regression

In the common nonparametric regression model Y_i=m(X_i)+sigma(X_i)epsilon_i we consider the problem of testing the hypothesis that the coefficient of the scale and location function is constant. The test is based on a comparison of the observations Y_i=\hat{sigma}(X_i) with their mean by a smoothed empirical process, where \hat{sigma} denotes the local linear estimate of the scale function. We show weak convergence of a centered version of this process to a Gaussian process under the null hypothesis and the alternative and use this result to construct a test for the hypothesis of a constant coefficient of variation in the nonparametric regression model. A small simulation study is also presented to investigate the finite sample properties of the new test. AMS Subject Classi cation: 62G10, 62F3

Censored quantile regression processes under dependence and penalization

We consider quantile regression processes from censored data under dependent data structures and derive a uniform Bahadur representation for those processes. We also consider cases where the dimension of the parameter in the quantile regression model is large. It is demonstrated that traditional penalization methods such as the adaptive lasso yield sub-optimal rates if the coe fficients of the quantile regression cross zero. New penalization techniques are introduced which are able to deal with speci c problems of censored data and yield estimates with an optimal rate. In contrast to most of the literature, the asymptotic analysis does not require the assumption of independent observations, but is based on rather weak assumptions, which are satis ed for many kinds of dependent data

Nonparametric comparison of quantile curves

A new test for comparing conditional quantile curves is proposed which is able to detect Pitman alternatives converging to the null hypothesis at the optimal rate. The basic idea of the test is to measure differences between the curves by a process of integrated non parametric estimates of the quantile curve. We prove weak convergence of this process to a Gaussian process and study the finite sample properties of a Kolmogorov-Smirnov test by means of a simulation study

The quantile process under random censoring

In this paper we discuss the asymptotical properties of quantile processes under random censoring. In contrast to most work in this area we prove weak convergence of an appropriately standardized quantile process under the assumption that the quantile regression model is only linear in the region, where the process is investigated. Additionally, we also discuss properties of the quantile process in sparse regression models including quantile processes obtained from the Lasso and adaptive Lasso. The results are derived by a combination of modern empirical process theory, classical martingale methods and a recent result of Kato (2009)

Bridge estimators and the adaptive Lasso under heteroscedasticity

In this paper we investigate penalized least squares methods in linear regression models with heteroscedastic error structure. It is demonstrated that the basic properties with respect to model selection and parameter estimation of bridge estimators, Lasso and adaptive Lasso do not change if the assumption of homoscedasticity is violated. However, these estimators do not have oracle properties in the sense o

Large Deviations for Random Matricial Moment Problems

We consider the moment space $\mathcal{M}_n^{K}$ corresponding to $p \times p$ complex matrix measures defined on $K$ ($K=[0,1]$ or K=\D). We endow this set with the uniform law. We are mainly interested in large deviations principles (LDP) when $n \rightarrow \infty$. First we fix an integer $k$ and study the vector of the first $k$ components of a random element of $\mathcal{M}_n^{K}$. We obtain a LDP in the set of $k$-arrays of $p\times p$ matrices. Then we lift a random element of $\mathcal{M}_n^{K}$ into a random measure and prove a LDP at the level of random measures. We end with a LDP on Carth\'eodory and Schur random functions. These last functions are well connected to the above random measure. In all these problems, we take advantage of the so-called canonical moments technique by introducing new (matricial) random variables that are independent and have explicit distributions.Comment: 34 page

A13K-0336: Airborne Multi-Wavelength High Spectral Resolution Lidar for Process Studies and Assessment of Future Satellite Remote Sensing Concepts

NASA Langley recently developed the world's first airborne multi-wavelength high spectral resolution lidar (HSRL). This lidar employs the HSRL technique at 355 and 532 nm to make independent, unambiguous retrievals of aerosol extinction and backscatter. It also employs the standard backscatter technique at 1064 nm and is polarization-sensitive at all three wavelengths. This instrument, dubbed HSRL-2 (the secondgeneration HSRL developed by NASA Langley), is a prototype for the lidar on NASA's planned Aerosols- Clouds-Ecosystems (ACE) mission. HSRL-2 completed its first science mission in July 2012, the Two-Column Aerosol Project (TCAP) conducted by the Department of Energy (DOE) in Hyannis, MA. TCAP presents an excellent opportunity to assess some of the remote sensing concepts planned for ACE: HSRL-2 was deployed on the Langley King Air aircraft with another ACE-relevant instrument, the NASA GISS Research Scanning Polarimeter (RSP), and flights were closely coordinated with the DOE's Gulfstream-1 aircraft, which deployed a variety of in situ aerosol and trace gas instruments and the new Spectrometer for Sky-Scanning, Sun-Tracking Atmospheric Research (4STAR). The DOE also deployed their Atmospheric Radiation Measurement Mobile Facility and their Mobile Aerosol Observing System at a ground site located on the northeastern coast of Cape Cod for this mission. In this presentation we focus on the capabilities, data products, and applications of the new HSRL-2 instrument. Data products include aerosol extinction, backscatter, depolarization, and optical depth; aerosol type identification; mixed layer depth; and rangeresolved aerosol microphysical parameters (e.g., effective radius, index of refraction, single scatter albedo, and concentration). Applications include radiative closure studies, studies of aerosol direct and indirect effects, investigations of aerosol-cloud interactions, assessment of chemical transport models, air quality studies, present (e.g., CALIPSO) and future (e.g., EarthCARE) satellite calibration/validation, and development/assessment of advanced retrieval techniques for future satellite applications (e.g., lidar+polarimeter retrievals of aerosol and cloud properties). We will also discuss the relevance of HSRL-2 measurement capabilities to the ACE remote sensing concept