22,215 research outputs found
Testing linear hypotheses in high-dimensional regressions
For a multivariate linear model, Wilk's likelihood ratio test (LRT)
constitutes one of the cornerstone tools. However, the computation of its
quantiles under the null or the alternative requires complex analytic
approximations and more importantly, these distributional approximations are
feasible only for moderate dimension of the dependent variable, say .
On the other hand, assuming that the data dimension as well as the number
of regression variables are fixed while the sample size grows, several
asymptotic approximations are proposed in the literature for Wilk's \bLa
including the widely used chi-square approximation. In this paper, we consider
necessary modifications to Wilk's test in a high-dimensional context,
specifically assuming a high data dimension and a large sample size .
Based on recent random matrix theory, the correction we propose to Wilk's test
is asymptotically Gaussian under the null and simulations demonstrate that the
corrected LRT has very satisfactory size and power, surely in the large and
large context, but also for moderately large data dimensions like or
. As a byproduct, we give a reason explaining why the standard chi-square
approximation fails for high-dimensional data. We also introduce a new
procedure for the classical multiple sample significance test in MANOVA which
is valid for high-dimensional data.Comment: Accepted 02/2012 for publication in "Statistics". 20 pages, 2 pages
and 2 table
Multipartite quantum correlation and entanglement in four-qubit pure states
Based on the quantitative complementarity relations, we analyze thoroughly
the properties of multipartite quantum correlations and entanglement in
four-qubit pure states. We find that, unlike the three-qubit case, the single
residual correlation, the genuine three- and four-qubit correlations are not
suited to quantify entanglement. More interestingly, from our qualitative and
numerical analysis, it is conjectured that the sum of all the residual
correlations may constitute a good measure for the total multipartite
entanglement in the system.Comment: 7 pages, 3 figue
SU(3) Family Gauge Symmetry and the Axion
We analyze the structure of a recently proposed effective field theory (EFT)
for the generation of quark and lepton mass ratios and mixing angles, based on
the spontaneous breaking of an SU(3) family gauge symmetry at a high scale F.
We classify the Yukawa operators necessary to seed the masses, making use of
the continuous global symmetries that they preserve. One global U(1), in
addition to baryon number and electroweak hypercharge, remains unbroken after
the inclusion of all operators required by standard-model-fermion
phenomenology. An associated vacuum symmetry insures the vanishing of the
first-family quark and charged-lepton masses in the absence of the family gauge
interaction. If this U(1) symmetry is taken to be exact in the EFT, broken
explicitly by only the QCD-induced anomaly, and if the breaking scale F is
taken to lie in the range 10 to 9 - 10 to 12 GeV, then the associated
Nambu-Goldstone boson is a potential QCD axion.Comment: References added and clarifications in Vacuum Structure sectio
Neutrinos and SU(3) Family Gauge Symmetry
We include the standard-model (SM) leptons in a recently proposed framework
for the generation of quark mass ratios and Cabibbo-Kobayashi-Maskawa (CKM)
mixing angles from an SU(3) family gauge interaction. The set of SM-singlet
scalar fields describing the spontaneous breaking is the same as employed for
the quark sector. The imposition at tree-level of the experimentally correct
Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix, in the form of a tri-bi
maximal structure, fixes several of the otherwise free parameters and renders
the model predictive. The normal hierarchy among the neutrino masses emerges
from this scheme.Comment: 9 pages, 3 tables; a comment added to clarify the effects of
additional Yukawa operators; final version in PR
Asymptotic properties of eigenmatrices of a large sample covariance matrix
Let where is a matrix
with i.i.d. complex standardized entries having finite fourth moments. Let
in which
and where
is the Mar\v{c}enko--Pastur law with parameter ; which
converges to a positive constant as , and and are unit vectors in ,
having indices and , ranging in a compact subset
of a finite-dimensional Euclidean space. In this paper, we prove that the
sequence converges weakly to a
-dimensional Gaussian process. This result provides further evidence in
support of the conjecture that the distribution of the eigenmatrix of is
asymptotically close to that of a Haar-distributed unitary matrix.Comment: Published in at http://dx.doi.org/10.1214/10-AAP748 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Multipartite entanglement in four-qubit cluster-class states
Based on quantitative complementarity relations (QCRs), we analyze the
multipartite correlations in four-qubit cluster-class states. It is proven
analytically that the average multipartite correlation is entanglement
monotone. Moreover, it is also shown that the mixed three-tangle is a
correlation measure compatible with the QCRs in this kind of quantum states.
More arrestingly, with the aid of the QCRs, a set of hierarchy entanglement
measures is obtained rigorously in the present system.Comment: 7 pages, 3 figs, version 3, some refs. are adde
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