Interiors of completely positive cones

A symmetric matrix $A$ is completely positive (CP) if there exists an entrywise nonnegative matrix $B$ such that $A = BB^T$. We characterize the interior of the CP cone. A semidefinite algorithm is proposed for checking interiors of the CP cone, and its properties are studied. A CP-decomposition of a matrix in Dickinson's form can be obtained if it is an interior of the CP cone. Some computational experiments are also presented

Star clusters in M33: updated UBVRI photometry, ages, metallicities, and masses

The photometric characterization of M33 star clusters is far from complete. In this paper, we present homogeneous $UBVRI$ photometry of 708 star clusters and cluster candidates in M33 based on archival images from the Local Group Galaxies Survey, which covers 0.8 deg$^2$ along the galaxy's major axis. Our photometry includes 387, 563, 616, 580, and 478 objects in the $UBVRI$ bands, respectively, of which 276, 405, 430, 457, and 363 do not have previously published $UBVRI$ photometry. Our photometry is consistent with previous measurements (where available) in all filters. We adopted Sloan Digital Sky Survey $ugriz$ photometry for complementary purposes, as well as Two Micron All-Sky Survey near-infrared $JHK$ photometry where available. We fitted the spectral-energy distributions of 671 star clusters and candidates to derive their ages, metallicities, and masses based on the updated {\sc parsec} simple stellar populations synthesis models. The results of our $\chi^2$ minimization routines show that only 205 of the 671 clusters ($31\%$) are older than 2 Gyr, which represents a much smaller fraction of the cluster population than that in M31 ($56\%$), suggesting that M33 is dominated by young star clusters ($<1$ Gyr). We investigate the mass distributions of the star clusters---both open and globular clusters---in M33, M31, the Milky Way, and the Large Magellanic Cloud. Their mean values are $\log(M_{\rm cl}/M_{\odot})=4.25$, 5.43, 2.72, and 4.18, respectively. The fraction of open to globular clusters is highest in the Milky Way and lowest in M31. Our comparisons of the cluster ages, masses, and metallicities show that our results are basically in agreement with previous studies (where objects in common are available); differences can be traced back to differences in the models adopted, the fitting methods used, and stochastic sampling effects.Comment: 32 pages, 12 figures, 2 tables, accepted for publication in ApJ

Hazard models with varying coefficients for multivariate failure time data

Statistical estimation and inference for marginal hazard models with varying coefficients for multivariate failure time data are important subjects in survival analysis. A local pseudo-partial likelihood procedure is proposed for estimating the unknown coefficient functions. A weighted average estimator is also proposed in an attempt to improve the efficiency of the estimator. The consistency and asymptotic normality of the proposed estimators are established and standard error formulas for the estimated coefficients are derived and empirically tested. To reduce the computational burden of the maximum local pseudo-partial likelihood estimator, a simple and useful one-step estimator is proposed. Statistical properties of the one-step estimator are established and simulation studies are conducted to compare the performance of the one-step estimator to that of the maximum local pseudo-partial likelihood estimator. The results show that the one-step estimator can save computational cost without compromising performance both asymptotically and empirically and that an optimal weighted average estimator is more efficient than the maximum local pseudo-partial likelihood estimator. A data set from the Busselton Population Health Surveys is analyzed to illustrate our proposed methodology.Comment: Published at http://dx.doi.org/10.1214/009053606000001145 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

The balanced Voronoi formulas for GL(n)

In this paper we show how the GL(N) Voronoi summation formula of [MiSc2] can be rewritten to incorporate hyper-Kloosterman sums of various dimensions on both sides. This generalizes a formula for GL(4) with ordinary Kloosterman sums on both sides that was considered by Xiaoqing Li and the first-named author, and later by the second-named author in [Zho]