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

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

The acute respiratory distress syndrome in 2013.

Acute lung injury and the acute respiratory distress syndrome are major causes of morbidity and mortality in critically ill patients. This review focuses on new developments in definitions, epidemiology, clinical and basic research, and promising new directions in treatment. There is new information about the potential contribution of environmental factors, especially exposure to cigarette smoke. Pathologic findings in ARDS have been limited to case reports of open lung biopsies and post-mortem studies but there is some new information from a recent pathology study relative to the frequency of diffuse alveolar damage and the severity of arterial hypoxemia. Further, therapy with lung-protective ventilation and fluid conservative protocol has improved outcomes, but several new trials are in progress to test several promising strategies

Limiting Government Predation Through Anonymous Banking: A Theory with Evidence from China

China's economic performance of the past two decades presents a puzzle for the economics of transition and development: Enormous private business incentives were unleashed that have fueled rapid economic growth despite the fact that China has had very weak "conventional institutions" (such as the rule of law and separation of powers) to constrain the government from arbitrary intrusion into economic activities. We argue that one mechanism that has limited the government's ability for predation and harassment is commitment through information decentralization, where the key institutiton is "anonymous banking," that is, a combination of the use of cash for transactions and the use of anonymous savings deposits. The government's incentive for such a mechanism comes form the increased quasi-fiscal revenues collected from the state banking system through "financial repression," a combination of controls on international capital flows with restrictions on domestic interest rates. The major features of China's economy concerning its fiscal decline, financial deepening, and the sectoral dual-track can be better understood using this analytical framework.http://deepblue.lib.umich.edu/bitstream/2027.42/39659/3/wp275.pd

A Multi-Task Theory of the State Enterprise Reform

During transition, maintaining employment and providing a social safety net to the unemployed are important to social stability, which in turn is crucial for the productivity of the whole economy. Because independent institutions for social safety are lacking and firms with strong profit incentives have little incentives to promote social stability due to its public good nature, state-owned enterprises (SOEs) are needed to continue their role in providing social welfare. Charged with the multi-tasks of efficient production as well as social welfare provision, SOEs continue to be given low profit incentives and consequently, their financial performance continues to be poor.http://deepblue.lib.umich.edu/bitstream/2027.42/39751/3/wp367.pd