Asymptotically minimax Bayes predictive densities

Given a random sample from a distribution with density function that depends on an unknown parameter $\theta$, we are interested in accurately estimating the true parametric density function at a future observation from the same distribution. The asymptotic risk of Bayes predictive density estimates with Kullback--Leibler loss function $D(f_{\theta}||{\hat{f}})=\int{f_{\theta} \log{(f_{\theta}/ hat{f})}}$ is used to examine various ways of choosing prior distributions; the principal type of choice studied is minimax. We seek asymptotically least favorable predictive densities for which the corresponding asymptotic risk is minimax. A result resembling Stein's paradox for estimating normal means by the maximum likelihood holds for the uniform prior in the multivariate location family case: when the dimensionality of the model is at least three, the Jeffreys prior is minimax, though inadmissible. The Jeffreys prior is both admissible and minimax for one- and two-dimensional location problems.Comment: Published at http://dx.doi.org/10.1214/009053606000000885 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Multiscale autocorrelation function: a new approach to anisotropy studies

We present a novel catalog-independent method, based on a scale dependent approach, to detect anisotropy signatures in the arrival direction distribution of the ultra highest energy cosmic rays (UHECR). The method provides a good discrimination power for both large and small data sets, even in presence of strong contaminating isotropic background. We present some applications to simulated data sets of events corresponding to plausible scenarios for charged particles detected by world-wide surface detector-based observatories, in the last decades.Comment: 18 pages, 9 figure

Objective prior for the number of degrees of freedom of a t distribution

In this paper, we construct an objective prior for the degrees of freedom of a t distribution, when the parameter is taken to be discrete. This parameter is typically problematic to estimate and a problem in objective Bayesian inference since improper priors lead to improper posteriors, whilst proper priors may dom- inate the data likelihood. We find an objective criterion, based on loss functions, instead of trying to define objective probabilities directly. Truncating the prior on the degrees of freedom is necessary, as the t distribution, above a certain number of degrees of freedom, becomes the normal distribution. The defined prior is tested in simulation scenarios, including linear regression with t-distributed errors, and on real data: the daily returns of the closing Dow Jones index over a period of 98 days