The general common nonnegative-definite and positive-definite solutions to the matrix equations AXA∗ = BB∗ and CXC∗ = DD∗

AbstractWe give necessary and sufficient conditions for the existence of a common nonnegative-definite (positive-definite) solution to the pair of matrix equations AXA∗ = BB∗ and CXC∗ = DD∗, and derive a representation of the general common nonnegative-definite (positive-definite) solution to these two equations when they have such common solutions. This paper can be viewed as a supplementary version of that derived by Young et al. [1] since Groβ [2] has given a counterexample to point out a mistake in their basic Theorem 1. The presented example shows the advantage of the proposed approach

Nonparametric Estimation of the Bivariate Recurrence Time Distribution

This paper considers statistical models in which two different types of events, such as the diagnosis of a disease and the remission of the disease, occur alternately over time and are observed subject to right censoring. We propose nonparametric estimators for the joint distribution of bivariate recurrence times and the marginal distribution of the first recurrence time. In general, the marginal distribution of the second recurrence time cannot be estimated due to an identifiability problem, but a conditional distribution of the second recurrence time can be estimated non-parametrically. In literature, statistical methods have been developed to estimate the joint distribution of bivariate recurrence times based on data of the first pair of censored bivariate recurrence times. These methods are efficient in the current model because recurrence times of higher orders are not used. Asymptotic properties of the estimators are established. Numerical studies demonstrate the estimator performs well with practical sample sizes. We apply the proposed method to a Denmark psychiatric case register data set for illustration of the methods and theory

Joint Modeling and Estimation for Recurrent Event Processes and Failure Time Data

Recurrent event data are commonly encountered in longitudinal follow-up studies related to biomedical science, econometrics, reliability, and demography. In many studies, recurrent events serve as important measurements for evaluating disease progression, health deterioration, or insurance risk. When analyzing recurrent event data, an independent censoring condition is typically required for the construction of statistical methods. Nevertheless, in some situations, the terminating time for observing recurrent events could be correlated with the recurrent event process and, as a result, the assumption of independent censoring is violated. In this paper, we consider joint modeling of a recurrent event process and a failure time in which a common subject-specific latent variable is used to model the association between the intensity of the recurrent event process and the hazard of the failure time. The proposed joint model is flexible in that no parametric assumptions on the distributions of censoring times and latent variables are made and, under the model, informative censoring is allowed for observing both the recurrent events and failure times. We propose a ‘borrow-strength estimation procedure’ by first estimating the value of the latent variable from recurrent event data, and next using the estimated value in the failure time model. Some interesting implications and trajectories of the proposed model will be presented. Properties of the regression parameter estimates and the estimated baseline cumulative hazard functions are also studied

Analyzing Panel Count Data with Informative Observation Times

In this paper, we study panel count data with informative observation times. We assume nonparametric and semiparametric proportional rate models for the underlying recurrent event process, where the form of the baseline rate function is left unspecified and a subject-specific frailty variable inflates or deflates the rate function multiplicatively. The proposed models allow the recurrent event processes and observation times to be correlated through their connections with the unobserved frailty; moreover, the distributions of both the frailty variable and observation times are considered as nuisance parameters. The baseline rate function and the regression parameters are estimated by maximizing a conditional likelihood function of observed event counts and solving estimation equations. Large sample properties of the proposed estimators are studied. Numerical studies demonstrate that the proposed estimation procedures perform well for moderate sample sizes. An application to a bladder tumor study is presented to illustrate the use of the proposed methods

Systematic study of elliptic flow parameter in the relativistic nuclear collisions at RHIC and LHC energies

We employed the new issue of a parton and hadron cascade model PACIAE 2.1 to systematically investigate the charged particle elliptic flow parameter $v_2$ in the relativistic nuclear collisions at RHIC and LHC energies. With randomly sampling the transverse momentum $x$ and $y$ components of the particles generated in string fragmentation on the circumference of an ellipse instead of circle originally, the calculated charged particle $v_2(\eta)$ and $v_2(p_T)$ fairly reproduce the corresponding experimental data in the Au+Au/Pb+Pb collisions at $\sqrt{s_{NN}}$=0.2/2.76 TeV. In addition, the charged particle $v_2(\eta)$ and $v_2(p_T)$ in the p+p collisions at $\sqrt s$=7 TeV as well as in the p+Au/p+Pb collisions at $\sqrt{s_{NN}}$=0.2/5.02 TeV are predicted.Comment: 7 pages, 5 figure

Kernel Estimation of Rate Function for Recurrent Event Data

Recurrent event data are largely characterized by the rate function but smoothing techniques for estimating the rate function have never been rigorously developed or studied in statistical literature. This paper considers the moment and least squares methods for estimating the rate function from recurrent event data. With an independent censoring assumption on the recurrent event process, we study statistical properties of the proposed estimators and propose bootstrap procedures for the bandwidth selection and for the approximation of confidence intervals in the estimation of the occurrence rate function. It is identified that the moment method without resmoothing via a smaller bandwidth will produce curve with nicks occurring at the censoring times, whereas there is no such problem with the least squares method. Furthermore, the asymptotic variance of the least squares estimator is shown to be smaller under regularity conditions. However, in the implementation of the bootstrap procedures, the moment method is computationally more efficient than the least squares method because the former approach uses condensed bootstrap data. The performance of the proposed procedures is studied through Monte Carlo simulations and an epidemiological example on intravenous drug users