Associate Professor, Department of Mathematical SciencesUniversity of Arkansas
Bayesian Analysis of Long Memory Time Series
In recent years there has been a growing interest for the statistical analysis of long memory processes, i.e., stationary processes with a spectral density presenting a pole at the origin. In this dissertation, I will present a Bayesian nonparametric approach to the problem. I propose a class of prior distributions on a family of spectral densities which properly includes a set of densities having a pole at the zero frequency. This allows to test for the presence of a long memory behaviour and to compare the prior and posterior probability of the data coming from a long memory process, in order to see how strongly the data support this hypothesis. The basic model is then extended to include a linear regression term for the mean of the process. Using Markov Chain Monte Carlo techniques, samples of the spectral density and the regression parameters from the posterior distribution can be obtained. With some modifications, the simulation scheme can be used to obtain also a sample from the predictive distribution of the future observations. The methodology described is applied to the analysis of the average temperature of the Southern Hemisphere over the last century.