Shape restricted regression with random Bernstein polynomials

Shape restricted regressions, including isotonic regression and concave regression as special cases, are studied using priors on Bernstein polynomials and Markov chain Monte Carlo methods. These priors have large supports, select only smooth functions, can easily incorporate geometric information into the prior, and can be generated without computational difficulty. Algorithms generating priors and posteriors are proposed, and simulation studies are conducted to illustrate the performance of this approach. Comparisons with the density-regression method of Dette et al. (2006) are included.Comment: Published at http://dx.doi.org/10.1214/074921707000000157 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Bayesian spectral modeling for multiple time series

We develop a novel Bayesian modeling approach to spectral density estimation for multiple time series. The log-periodogram distribution for each series is modeled as a mixture of Gaussian distributions with frequency-dependent weights and mean functions. The implied model for the log-spectral density is a mixture of linear mean functions with frequency-dependent weights. The mixture weights are built through successive differences of a logit-normal distribution function with frequency-dependent parameters. Building from the construction for a single spectral density, we develop a hierarchical extension for multiple time series. Specifically, we set the mean functions to be common to all spectral densities and make the weights specific to the time series through the parameters of the logit-normal distribution. In addition to accommodating flexible spectral density shapes, a practically important feature of the proposed formulation is that it allows for ready posterior simulation through a Gibbs sampler with closed form full conditional distributions for all model parameters. The modeling approach is illustrated with simulated datasets, and used for spectral analysis of multichannel electroencephalographic recordings (EEGs), which provides a key motivating application for the proposed methodology

Bayesian methods for function estimation

Key words and phrases. Consistency, convergence rate, Dirichlet process, density estimation, Markov chain Monte-Carlo, posterior distribution, regression function, spectral density, transition density.

Recent advances in directional statistics

Mainstream statistical methodology is generally applicable to data observed in Euclidean space. There are, however, numerous contexts of considerable scientific interest in which the natural supports for the data under consideration are Riemannian manifolds like the unit circle, torus, sphere and their extensions. Typically, such data can be represented using one or more directions, and directional statistics is the branch of statistics that deals with their analysis. In this paper we provide a review of the many recent developments in the field since the publication of Mardia and Jupp (1999), still the most comprehensive text on directional statistics. Many of those developments have been stimulated by interesting applications in fields as diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics, image analysis, text mining, environmetrics, and machine learning. We begin by considering developments for the exploratory analysis of directional data before progressing to distributional models, general approaches to inference, hypothesis testing, regression, nonparametric curve estimation, methods for dimension reduction, classification and clustering, and the modelling of time series, spatial and spatio-temporal data. An overview of currently available software for analysing directional data is also provided, and potential future developments discussed.Comment: 61 page

Non-parametric methods for circular-circular and circular-linear

We present a non-parametric approach for the estimation of the bivariate distribution of two circular variables and the modelling of the joint distribution of a circular and a linear variable. We combine nonparametric estimates of the marginal densities of the circular and linear components with the use of class of nonparametric copulas, known as empirical Bernstein copulas, to model the dependence structure. We derive the necessary conditions to obtain continuous distributions defined on the cylinder for the circular-linear model and on the torus for the circular-circular model. We illustrate these two approaches with two sets of real environmental dataBernstein polynomials, Circular distributions, Circular-Circular data, Circular-linear data, Copulas, Non-parametric estimation