Quantifying and Estimating the Predictive Accuracy for Censored Time-to-Event Data with Competing Risks

This paper focuses on quantifying and estimating the predictive accuracy of prognostic models for time-to-event outcomes with competing events. We consider the time-dependent discrimination and calibration metrics, including the receiver operating characteristics curve and the Brier score, in the context of competing risks. To address censoring, we propose a unified nonparametric estimation framework for both discrimination and calibration measures, by weighting the censored subjects with the conditional probability of the event of interest given the observed data. We demonstrate through simulations that the proposed estimator is unbiased, efficient and robust against model misspecification in comparison to other methods published in the literature. In addition, the proposed method can be extended to time-dependent predictive accuracy metrics constructed from a general class of loss functions. We apply the methodology to a data set from the African American Study of Kidney Disease and Hypertension to evaluate the predictive accuracy of a prognostic risk score in predicting end-stage renal disease (ESRD), accounting for the competing risk of pre-ESRD death

Statistical and Computational Limits for Sparse Matrix Detection

This paper investigates the fundamental limits for detecting a high-dimensional sparse matrix contaminated by white Gaussian noise from both the statistical and computational perspectives. We consider $p\times p$ matrices whose rows and columns are individually $k$-sparse. We provide a tight characterization of the statistical and computational limits for sparse matrix detection, which precisely describe when achieving optimal detection is easy, hard, or impossible, respectively. Although the sparse matrices considered in this paper have no apparent submatrix structure and the corresponding estimation problem has no computational issue at all, the detection problem has a surprising computational barrier when the sparsity level $k$ exceeds the cubic root of the matrix size $p$: attaining the optimal detection boundary is computationally at least as hard as solving the planted clique problem. The same statistical and computational limits also hold in the sparse covariance matrix model, where each variable is correlated with at most $k$ others. A key step in the construction of the statistically optimal test is a structural property for sparse matrices, which can be of independent interest

Hawking Radiation of Dirac Particles in a Variable-mass Kerr Black Hole

Hawking effect of Dirac particles in a variable-mass Kerr space-time is investigated by using method of the generalized tortoise coordinate transformation. The location and the temperature of event horizon of the non-stationary Kerr black hole are derived. It is shown that the temperature and the shape of event horizon depend not only on the time but also on the polar angle. However, our results demonstrate that the Fermi-Dirac spectrum displays a new spin-rotation effect which is absent from that of Bose-Einstein distribution.Comment: 6 pages, revtex (12pt), no figure. Chin. Phys. Lett. 18 (2001) 485 (in press

Hawking Radiation of a Non-stationary Kerr-Newman Black Hole: Spin-Rotation Coupling Effect

Hawking evaporation of Klein-Gordon and Dirac particles in a non-stationary Kerr-Newman space-time is investigated by using a method of generalized tortoise coordinate transformation. The location and the temperature of the event horizon of a non-stationary Kerr-Newman black hole are derived. It is shown that the temperature and the shape of the event horizon depend not only on the time but also on the angle. However, the Fermionic spectrum of Dirac particles displays a new spin-rotation coupling effect which is absent from that of Bosonic distribution of scalar particles. The character of this effect is its obvious dependence on different helicity states of particles spin-1/2. PACS numbers: 04.70.Dy, 97.60.LfComment: 12 pages, revtex, no figure, to appear in Gen. Rel. Grav. 34 (2002) No.

Hawking Radiation of Weyl Neutrinos in a Rectilinearly Non-uniformly Accelerating Kinnersley Black Hole

Quantum thermal effect of Weyl neutrinos in a rectilinearly non-uniformly accelerating Kinnersley black hole is investigated by using the generalized tortoise coordinate transformation. The equation that determines the location, the Hawking temperature of the event horizon and the thermal radiation spectrum of neutrinos are derived. Our results show that the location and the temperature of the event horizon depend not only on the time but also on the angle.Comment: 9 pages, no figure, Latex 2.09, accepted for Chinese Physics Vol. 11, No. 7 (2002

Generalized Laws of Black Hole Thermodynamics and Quantum Conservation Laws on Hawking Radiation Process

Four classical laws of black hole thermodynamics are extended from exterior (event) horizon to interior (Cauchy) horizon. Especially, the first law of classical thermodynamics for Kerr-Newman black hole (KNBH) is generalized to those in quantum form. Then five quantum conservation laws on the KNBH evaporation effect are derived in virtue of thermodynamical equilibrium conditions. As a by-product, Bekenstein-Hawking's relation $S=A/4$ is exactly recovered.Comment: Latex, 8 pages, no figur

No New Quantum Thermal Effect of Dirac Particles in a Charged Vaidya - de Sitter Black Hole

It is shown that Hawking radiation of Dirac particles does not exist for $P_1, Q_2$ components but for $P_2, Q_1$ components in a charged Vaidya - de Sitter black hole. Both the location and the temperature of the event horizon change with time. The thermal radiation spectrum of Dirac particles is the same as that of Klein-Gordon particles. Our result demonstrates that there is no new quantum effect in the thermal radiation of Dirac particles in any spherically symmetry black holes.Comment: 12pt revtex, 10 pages, no figure, accepted for IL Nuovo Cimento

Four Quantum Conservation Laws on Black Hole Equilibrium Radiation Process and Quantum Black Hole Entropy

The classical first law of thermodynamic for Kerr-Newmann black hole (KNBH) is generalized to that in quantum form on event horizon. Then four quantum conservation laws on the KNBH equilibrium radiation process are derived, and Bekenstein-Hawking's relation S=A/4 is recovered. It can be argued that the classical entropy of black hole arise from the quantum entropy of field quanta or quasi-particles inside the hole.Comment: 10 Pages, in Latex, no figur

Four Quantum Conservation Laws for Black Hole Stationary Equilibrium Radiation Processes

The classical first law of thermodynamics for a Kerr-Newman black hole (KNBH) is generalized to a law in quantum form on the event horizon. Then four quantum conservation laws on the KNBH equilibrium radiation process are derived. The Bekenstein-Hawking relation ${\cal{S}}={\cal{A}}/4$ is exactly established. It can be inferred that the classical entropy of black hole arises from the quantum entropy of field quanta or quasi-particles inside the hole.Comment: 7 pages, no figure, Revtex in 12p

Addendum: Hawking Radiation of Photons in a Variable-mass Kerr Black Hole

Hawking evaporation of photons in a variable-mass Kerr space-time is investigated by using a method of the generalized tortoise coordinate transformation. The blackbody radiant spectrum of photons displays a new spin-rotation coupling effect obviously dependent on different helicity states of photons.Comment: 8 pages, no figures, Latex(use kluwer.cls), to appear in Gen. Rel. Grav. 34 (2002) No.