2,428 research outputs found
A martingale-transform goodness-of-fit test for the form of the conditional variance
In the common nonparametric regression model the problem of testing for a
specific parametric form of the variance function is considered. Recently Dette
and Hetzler (2008) proposed a test statistic, which is based on an empirical
process of pseudo residuals. The process converges weakly to a Gaussian process
with a complicated covariance kernel depending on the data generating process.
In the present paper we consider a standardized version of this process and
propose a martingale transform to obtain asymptotically distribution free tests
for the corresponding Kolmogorov-Smirnov and Cram\'{e}r-von-Mises functionals.
The finite sample properties of the proposed tests are investigated by means of
a simulation study.Comment: 24 pages
Some asymptotic properties of the spectrum of the Jacobi ensemble
For the random eigenvalues with density corresponding to the Jacobi ensemble
a strong uniform approximation by the roots of the Jacobi polynomials is
derived if the parameters depend on and .
Roughly speaking, the eigenvalues can be uniformly approximated by roots of
Jacobi polynomials with parameters , where
the error is of order . These results are used to
investigate the asymptotic properties of the corresponding spectral
distribution if and the parameters and vary with
. We also discuss further applications in the context of multivariate random
-matrices.Comment: 20 pages, 2 figure
Moderate deviations for the eigenvalue counting function of Wigner matrices
We establish a moderate deviation principle (MDP) for the number of
eigenvalues of a Wigner matrix in an interval. The proof relies on fine
asymptotics of the variance of the eigenvalue counting function of GUE matrices
due to Gustavsson. The extension to large families of Wigner matrices is based
on the Tao and Vu Four Moment Theorem and applies localization results by
Erd\"os, Yau and Yin. Moreover we investigate families of covariance matrices
as well.Comment: 20 page
Optimal experimental designs for inverse quadratic regression models
In this paper optimal experimental designs for inverse quadratic regression
models are determined. We consider two different parameterizations of the model
and investigate local optimal designs with respect to the -, - and
-criteria, which reflect various aspects of the precision of the maximum
likelihood estimator for the parameters in inverse quadratic regression models.
In particular it is demonstrated that for a sufficiently large design space
geometric allocation rules are optimal with respect to many optimality
criteria. Moreover, in numerous cases the designs with respect to the different
criteria are supported at the same points. Finally, the efficiencies of
different optimal designs with respect to various optimality criteria are
studied, and the efficiency of some commonly used designs are investigated.Comment: 24 page
Optimal discrimination designs
We consider the problem of constructing optimal designs for model
discrimination between competing regression models. Various new properties of
optimal designs with respect to the popular -optimality criterion are
derived, which in many circumstances allow an explicit determination of
-optimal designs. It is also demonstrated, that in nested linear models the
number of support points of -optimal designs is usually too small to
estimate all parameters in the extended model. In many cases -optimal
designs are usually not unique, and in this situation we give a
characterization of all -optimal designs. Finally, -optimal designs are
compared with optimal discriminating designs with respect to alternative
criteria by means of a small simulation study.Comment: Published in at http://dx.doi.org/10.1214/08-AOS635 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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
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