The Bernstein-Von Mises Theorem in Semiparametric Competing Risks Models

Semiparametric Bayesian models are nowadays a popular tool in survival analysis. An important area of research concerns the investigation of frequentist properties of these models. In this paper, a Bernstein-von Mises theorem is derived for semiparametric Bayesian models of competing risks data. The cause-specific hazard is taken as the product of the conditional probability of a failure type and the overall hazard rate. We model the conditional probability as a smooth function of time and leave the cumulative overall hazard unspecified. A prior distribution is defined on the joint parameter space, which includes a beta process prior for the cumulative overall hazard. We show that the posterior distribution for any differentiable functional of interest is asymptotically equivalent to the sampling distribution derived from maximum likelihood estimation. A simulation study is provided to illustrate the coverage properties of credible intervals on cumulative incidence functions.Bayesian nonparametrics, Bernstein-von Mises theorem, beta process, competing risks, conditional probability of a failure type, semiparametric inference.

Alternative Descriptions in Quaternionic Quantum Mechanics

We characterize the quasianti-Hermitian quaternionic operators in QQM by means of their spectra; moreover, we state a necessary and sufficient condition for a set of quasianti-Hermitian quaternionic operators to be anti-Hermitian with respect to a uniquely defined positive scalar product in a infinite dimensional (right) quaternionic Hilbert space. According to such results we obtain two alternative descriptions of a quantum optical physical system, in the realm of quaternionic quantum mechanics, while no alternative can exist in complex quantum mechanics, and we discuss some differences between them.Comment: 16 page