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 $p\le 20$. On the other hand, assuming that the data dimension $p$ as well as the number $q$ of regression variables are fixed while the sample size $n$ 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 $p$ and a large sample size $n$. 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 $p$ and large $n$ context, but also for moderately large data dimensions like $p=30$ or $p=50$. 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 $S_n=\frac{1}{n}X_nX_n^*$ where $X_n=\{X_{ij}\}$ is a $p\times n$ matrix with i.i.d. complex standardized entries having finite fourth moments. Let $Y_n(\mathbf {t}_1,\mathbf {t}_2,\sigma)=\sqrt{p}({\mathbf {x}}_n(\mathbf {t}_1)^*(S_n+\sigma I)^{-1}{\mathbf {x}}_n(\mathbf {t}_2)-{\mathbf {x}}_n(\mathbf {t}_1)^*{\mathbf {x}}_n(\mathbf {t}_2)m_n(\sigma))$ in which $\sigma>0$ and $m_n(\sigma)=\int\frac{dF_{y_n}(x)}{x+\sigma}$ where $F_{y_n}(x)$ is the Mar\v{c}enko--Pastur law with parameter $y_n=p/n$; which converges to a positive constant as $n\to\infty$, and ${\mathbf {x}}_n(\mathbf {t}_1)$ and ${\mathbf {x}}_n(\mathbf {t}_2)$ are unit vectors in ${\Bbb{C}}^p$, having indices $\mathbf {t}_1$ and $\mathbf {t}_2$, ranging in a compact subset of a finite-dimensional Euclidean space. In this paper, we prove that the sequence $Y_n(\mathbf {t}_1,\mathbf {t}_2,\sigma)$ converges weakly to a $(2m+1)$-dimensional Gaussian process. This result provides further evidence in support of the conjecture that the distribution of the eigenmatrix of $S_n$ 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 $E_{ms}$ 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