Mixed Marginal Copula Modeling

This article extends the literature on copulas with discrete or continuous marginals to the case where some of the marginals are a mixture of discrete and continuous components. We do so by carefully defining the likelihood as the density of the observations with respect to a mixed measure. The treatment is quite general, although we focus focus on mixtures of Gaussian and Archimedean copulas. The inference is Bayesian with the estimation carried out by Markov chain Monte Carlo. We illustrate the methodology and algorithms by applying them to estimate a multivariate income dynamics model.Comment: 46 pages, 8 tables and 4 figure

Approximating multivariate posterior distribution functions from Monte Carlo samples for sequential Bayesian inference

An important feature of Bayesian statistics is the opportunity to do sequential inference: the posterior distribution obtained after seeing a dataset can be used as prior for a second inference. However, when Monte Carlo sampling methods are used for inference, we only have a set of samples from the posterior distribution. To do sequential inference, we then either have to evaluate the second posterior at only these locations and reweight the samples accordingly, or we can estimate a functional description of the posterior probability distribution from the samples and use that as prior for the second inference. Here, we investigated to what extent we can obtain an accurate joint posterior from two datasets if the inference is done sequentially rather than jointly, under the condition that each inference step is done using Monte Carlo sampling. To test this, we evaluated the accuracy of kernel density estimates, Gaussian mixtures, vine copulas and Gaussian processes in approximating posterior distributions, and then tested whether these approximations can be used in sequential inference. In low dimensionality, Gaussian processes are more accurate, whereas in higher dimensionality Gaussian mixtures or vine copulas perform better. In our test cases, posterior approximations are preferable over direct sample reweighting, although joint inference is still preferable over sequential inference. Since the performance is case-specific, we provide an R package mvdens with a unified interface for the density approximation methods

Modeling for seasonal marked point processes: An analysis of evolving hurricane occurrences

Seasonal point processes refer to stochastic models for random events which are only observed in a given season. We develop nonparametric Bayesian methodology to study the dynamic evolution of a seasonal marked point process intensity. We assume the point process is a nonhomogeneous Poisson process and propose a nonparametric mixture of beta densities to model dynamically evolving temporal Poisson process intensities. Dependence structure is built through a dependent Dirichlet process prior for the seasonally-varying mixing distributions. We extend the nonparametric model to incorporate time-varying marks, resulting in flexible inference for both the seasonal point process intensity and for the conditional mark distribution. The motivating application involves the analysis of hurricane landfalls with reported damages along the U.S. Gulf and Atlantic coasts from 1900 to 2010. We focus on studying the evolution of the intensity of the process of hurricane landfall occurrences, and the respective maximum wind speed and associated damages. Our results indicate an increase in the number of hurricane landfall occurrences and a decrease in the median maximum wind speed at the peak of the season. Introducing standardized damage as a mark, such that reported damages are comparable both in time and space, we find that there is no significant rising trend in hurricane damages over time.Comment: Published at http://dx.doi.org/10.1214/14-AOAS796 in the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org