Matrix measures, random moments and Gaussian ensembles

We consider the moment space $\mathcal{M}_n$ corresponding to $p \times p$ real or complex matrix measures defined on the interval $[0,1]$. The asymptotic properties of the first $k$ components of a uniformly distributed vector $(S_{1,n}, ..., S_{n,n})^* \sim \mathcal{U} (\mathcal{M}_n)$ are studied if $n \to \infty$. In particular, it is shown that an appropriately centered and standardized version of the vector $(S_{1,n}, ..., S_{k,n})^*$ converges weakly to a vector of $k$ independent $p \times p$ Gaussian ensembles. For the proof of our results we use some new relations between ordinary moments and canonical moments of matrix measures which are of their own interest. In particular, it is shown that the first $k$ canonical moments corresponding to the uniform distribution on the real or complex moment space $\mathcal{M}_n$ are independent multivariate Beta distributed random variables and that each of these random variables converge in distribution (if the parameters converge to infinity) to the Gaussian orthogonal ensemble or to the Gaussian unitary ensemble, respectively.Comment: 25 page

Distributions on unbounded moment spaces and random moment sequences

In this paper we define distributions on moment spaces corresponding to measures on the real line with an unbounded support. We identify these distributions as limiting distributions of random moment vectors defined on compact moment spaces and as distributions corresponding to random spectral measures associated with the Jacobi, Laguerre and Hermite ensemble from random matrix theory. For random vectors on the unbounded moment spaces we prove a central limit theorem where the centering vectors correspond to the moments of the Marchenko-Pastur distribution and Wigner's semi-circle law.Comment: Published in at http://dx.doi.org/10.1214/11-AOP693 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Bootstrap tests for the error distribution in linear and nonparametric regression models

In this paper we investigate several tests for the hypothesis of a parametric form of the error distribution in the common linear and nonparametric regression model, which are based on empirical processes of residuals. It is well known that tests in this context are not asymptotically distribution-free and the parametric bootstrap is applied to deal with this problem. The performance of the resulting bootstrap test is investigated from an asymptotic point of view and by means of a simulation study. The results demonstrate that even for moderate sample sizes the parametric bootstrap provides a reliable and easy accessible solution to the problem of goodness-of-fit testing of assumptions regarding the error distribution in linear and nonparametric regression models. --goodness-of-fit,residual process,parametric bootstrap,linear model,analysis of variance,M-estimation,nonparametric regression

A note on testing symmetry of the error distribution in linear regression models

In the classical linear regression model the problem of testing for symmetry of the error distribution is considered. The test statistic is a functional of the difference between the two empirical distribution functions of the estimated residuals and their counterparts with opposite signs. The weak convergence of the difference process to a Gaussian process is established. The covariance structure of this process depends heavily on the density of the error distribution, and for this reason the performance of a symmetric wild bootstrap procedure is discussed in asymptotic theory and by means of a simulation study. --M-estimation,goodness-of-fit tests,testing for symmetry,empirical process of residuals,linear model

Some asymptotic properties of the spectrum of the Jacobi ensemble

For the random eigenvalues with density corresponding to the Jacobi ensemble $$c \cdot \prod_{i < j} | \lambda_i - \lambda_j |^\beta \prod^n_{i=1} (2 - \lambda_i)^a (2 + \lambda_i)^b I_{(-2,2)} (\lambda_i)$$ $(a, b > -1, \beta > 0)$ a strong uniform approximation by the roots of the Jacobi polynomials is derived if the parameters $a, b,$ $\beta$ depend on $n$ and $n \to \infty$. Roughly speaking, the eigenvalues can be uniformly approximated by roots of Jacobi polynomials with parameters $((2a+2)/\beta -1, (2b+2)/\beta-1)$, where the error is of order $\{\log n/(a+b) \}^{1/4}$. These results are used to investigate the asymptotic properties of the corresponding spectral distribution if $n \to \infty$ and the parameters $a, b$ and $\beta$ vary with $n$. We also discuss further applications in the context of multivariate random $F$-matrices.Comment: 20 pages, 2 figure

Bootstrap Tests for the Error Distribution in Linear and Nonparametric Regression Models

In this paper we investigate several tests for the hypothesis of a parametric form of the error distribution in the common linear and nonparametric regression model, which are based on empirical processes of residuals. It is well known that tests in this context are not asymptotically distribution-free and the parametric bootstrap is applied to deal with this problem. The performance of the resulting bootstrap test is investigated from an asymptotic point of view and by means of a simulation study. The results demonstrate that even for moderate sample sizes the parametric bootstrap provides a reliable and easy accessible solution to the problem of goodness-of-fit testing of assumptions regarding the error distribution in linear and nonparametric regression models

A Note on Testing Symmetry of the Error Distribution in Linear Regression Models

In the classical linear regression model the problem of testing for symmetry of the error distribution is considered. The test statistic is a functional of the difference between the two empirical distribution functions of the estimated residuals and their counterparts with opposite signs. The weak convergence of the difference process to a Gaussian process is established. The covariance structure of this process depends heavily on the density of the error distribution, and for this reason the performance of a symmetric wild bootstrap procedure is discussed in asymptotic theory and by means of a simulation study

Random block matrices generalizing the classical Jacobi and Laguerre ensembles

AbstractIn this paper we consider random block matrices which generalize the classical Laguerre ensemble and the Jacobi ensemble. We show that the random eigenvalues of the matrices can be uniformly approximated by the zeros of matrix orthogonal polynomials and obtain a rate for the maximum difference between the eigenvalues and the zeros. This relation between the random block matrices and matrix orthogonal polynomials allows a derivation of the asymptotic spectral distribution of the matrices

CT radiomics to predict Deauville score 4 positive and negative Hodgkin lymphoma manifestations

18F-FDG-PET/CT is standard to assess response in Hodgkin lymphoma by quantifying metabolic activity with the Deauville score. PET/CT, however, is time-consuming, cost-extensive, linked to high radiation and has a low availability. As an alternative, we investigated radiomics from non-contrast-enhanced computed tomography (NECT) scans. 75 PET/CT examinations of 43 patients on two different scanners were included. Target lesions were classified as Deauville score 4 positive (DS4+) or negative (DS4-) based on their SUVpeak and then segmented in NECT images. From these segmentations, 107 features were extracted with PyRadiomics. All further statistical analyses were then performed scanner-wise: differences between DS4+ and DS4- manifestations were assessed with the Mann-Whitney-U-test and single feature performances with the ROC-analysis. To further verify the reliability of the results, the number of features was reduced using different techniques. The feature median showed a high sensitivity for DS4+ manifestations on both scanners (scanner A: 0.91, scanner B: 0.85). It furthermore was the only feature that remained in both datasets after applying different feature reduction techniques. The feature median from NECT concordantly has a high sensitivity for DS4+ Hodgkin manifestations on two different scanners and thus could provide a surrogate for increased metabolic activity in PET/CT