Adaptive confidence intervals for regression functions under shape constraints

Adaptive confidence intervals for regression functions are constructed under shape constraints of monotonicity and convexity. A natural benchmark is established for the minimum expected length of confidence intervals at a given function in terms of an analytic quantity, the local modulus of continuity. This bound depends not only on the function but also the assumed function class. These benchmarks show that the constructed confidence intervals have near minimum expected length for each individual function, while maintaining a given coverage probability for functions within the class. Such adaptivity is much stronger than adaptive minimaxity over a collection of large parameter spaces.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1068 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Two-Sample Covariance Matrix Testing and Support Recovery

This paper proposes a new test for testing the equality of two covariance matrices Σ1 and Σ2 in the high-dimensional setting and investigates its theoretical and numerical properties. The limiting null distribution of the test statistic is derived. The test is shown to enjoy certain optimality and to be especially powerful against sparse alternatives. The simulation results show that the test significantly outperforms the existing methods both in terms of size and power. Analysis of prostate cancer datasets is carried out to demonstrate the application of the testing procedures. When the null hypothesis of equal covariance matrices is rejected, it is often of significant interest to further investigate in which way they differ. Motivated by applications in genomics, we also consider two related problems, recovering the support of Σ1 − Σ2 and testing the equality of the two covariance matrices row by row. New testing procedures are introduced and their properties are studied. Applications to gene selection is also discussed

Plasmons and screening in monolayer and multilayer black phosphorus

Black phosphorus exhibits a high degree of band anisotropy. However, we found that its in-plane static screening remains relatively isotropic for momenta relevant to elastic long-range scattering processes. On the other hand, the collective electronic excitations in the system exhibit a strong anisotropy. Band non-parabolicity leads to a plasmon frequency which scales as $n^{\beta}$, where $n$ is the carrier concentration, and $\beta<\tfrac{1}{2}$. Screening and charge distribution in the out-of-plane direction are also studied using a non-linear Thomas-Fermi model

Testing Differential Networks with Applications to Detection of Gene-Gene Interactions

Model organisms and human studies have led to increasing empirical evidence that interactions among genes contribute broadly to genetic variation of complex traits. In the presence of gene-by-gene interactions, the dimensionality of the feature space becomes extremely high relative to the sample size. This imposes a significant methodological challenge in identifying gene-by-gene interactions. In the present paper, through a Gaussian graphical model framework, we translate the problem of identifying gene-by-gene interactions associated with a binary trai

Locally Adaptive Algorithms for Multiple Testing with Network Structure, with Application to Genome-Wide Association Studies

Linkage analysis has provided valuable insights to the GWAS studies, particularly in revealing that SNPs in linkage disequilibrium (LD) can jointly influence disease phenotypes. However, the potential of LD network data has often been overlooked or underutilized in the literature. In this paper, we propose a locally adaptive structure learning algorithm (LASLA) that provides a principled and generic framework for incorporating network data or multiple samples of auxiliary data from related source domains; possibly in different dimensions/structures and from diverse populations. LASLA employs a $p$-value weighting approach, utilizing structural insights to assign data-driven weights to individual test points. Theoretical analysis shows that LASLA can asymptotically control FDR with independent or weakly dependent primary statistics, and achieve higher power when the network data is informative. Efficiency again of LASLA is illustrated through various synthetic experiments and an application to T2D-associated SNP identification.Comment: 33 pages, 7 figure