No Arabic abstract
Rao-Blackwellization is a notion often occurring in the MCMC literature, with possibly different meanings and connections with the original Rao--Blackwell theorem (Rao, 1945 and Blackwell,1947), including a reduction of the variance of the resulting Monte Carlo approximations. This survey reviews some of the meanings of the term.
In this paper, we study the geometry induced by the Fisher-Rao metric on the parameter space of Dirichlet distributions. We show that this space is geodesically complete and has everywhere negative sectional curvature. An important consequence of this negative curvature for applications is that the Fr{e}chet mean of a set of Dirichlet distributions is uniquely defined in this geometry.
Sufficient statistics are derived for the population size and parameters of commonly used closed population mark-recapture models. Rao-Blackwellization details for improving estimators that are not functions of the statistics are presented. As Rao-Blackwellization entails enumerating all sample reorderings consistent with the sufficient statistic, Markov chain Monte Carlo resampling procedures are provided to approximate the computationally intensive estimators. Simulation studies demonstrate that significant improvements can be made with the strategy. Supplementary materials for this article are available online.
We gather several results on the eigenvalues of the spatial sign covariance matrix of an elliptical distribution. It is shown that the eigenvalues are a one-to-one function of the eigenvalues of the shape matrix and that they are closer together than the latter. We further provide a one-dimensional integral representation of the eigenvalues, which facilitates their numerical computation.
Among Monte Carlo techniques, the importance sampling requires fine tuning of a proposal distribution, which is now fluently resolved through iterative schemes. The Adaptive Multiple Importance Sampling (AMIS) of Cornuet et al. (2012) provides a significant improvement in stability and effective sample size due to the introduction of a recycling procedure. However, the consistency of the AMIS estimator remains largely open. In this work we prove the convergence of the AMIS, at a cost of a slight modification in the learning process. Contrary to Douc et al. (2007a), results are obtained here in the asymptotic regime where the number of iterations is going to infinity while the number of drawings per iteration is a fixed, but growing sequence of integers. Hence some of the results shed new light on adaptive population Monte Carlo algorithms in that last regime.
We consider the problem of computing the joint distribution of order statistics of stochastically independent random variables in one- and two-group models. While recursive formulas for evaluating the joint cumulative distribution function of such order statistics exist in the literature for a longer time, their numerical implementation remains a challenging task. We tackle this task by presenting novel generalizations of known recursions which we utilize to obtain exact results (calculated in rational arithmetic) as well as faithfully rounded results. Finally, some applications in stepwise multiple hypothesis testing are discussed.