14,579 research outputs found
Testing the Sphericity of a covariance matrix when the dimension is much larger than the sample size
This paper focuses on the prominent sphericity test when the dimension is
much lager than sample size . The classical likelihood ratio test(LRT) is no
longer applicable when . Therefore a Quasi-LRT is proposed and
asymptotic distribution of the test statistic under the null when
is well established in this paper.
Meanwhile, John's test has been found to possess the powerful {\it
dimension-proof} property, which keeps exactly the same limiting distribution
under the null with any -asymptotic, i.e. ,
. All asymptotic results are derived for general population
with finite fourth order moment. Numerical experiments are implemented for
comparison
Restricted Flows and the Soliton Equation with Self-Consistent Sources
The KdV equation is used as an example to illustrate the relation between the
restricted flows and the soliton equation with self-consistent sources.
Inspired by the results on the Backlund transformation for the restricted flows
(by V.B. Kuznetsov et al.), we constructed two types of Darboux transformations
for the KdV equation with self-consistent sources (KdVES). These Darboux
transformations are used to get some explicit solutions of the KdVES, which
include soliton, rational, positon, and negaton solutions.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
On singular value distribution of large dimensional auto-covariance matrices
Let be a sequence of independent dimensional
random vectors and a given integer. From a sample
of the
sequence, the so-called lag auto-covariance matrix is
. When the
dimension is large compared to the sample size , this paper establishes
the limit of the singular value distribution of assuming that and
grow to infinity proportionally and the sequence satisfies a Lindeberg
condition on fourth order moments. Compared to existing asymptotic results on
sample covariance matrices developed in random matrix theory, the case of an
auto-covariance matrix is much more involved due to the fact that the summands
are dependent and the matrix is not symmetric. Several new techniques
are introduced for the derivation of the main theorem
The generalized Kupershmidt deformation for constructing new integrable systems from integrable bi-Hamiltonian systems
Based on the Kupershmidt deformation for any integrable bi-Hamiltonian
systems presented in [4], we propose the generalized Kupershmidt deformation to
construct new systems from integrable bi-Hamiltonian systems, which provides a
nonholonomic perturbation of the bi-Hamiltonian systems. The generalized
Kupershmidt deformation is conjectured to preserve integrability. The
conjecture is verified in a few representative cases: KdV equation, Boussinesq
equation, Jaulent-Miodek equation and Camassa-Holm equation. For these specific
cases, we present a general procedure to convert the generalized Kupershmidt
deformation into the integrable Rosochatius deformation of soliton equation
with self-consistent sources, then to transform it into a -type
bi-Hamiltonian system. By using this generalized Kupershmidt deformation some
new integrable systems are derived. In fact, this generalized Kupershmidt
deformation also provides a new method to construct the integrable Rosochatius
deformation of soliton equation with self-consistent sources.Comment: 21 pages, to appear in Journal of Mathematical Physic
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