Bayesian inference for stable processes

The problem of Bayesian inference for univariate and multivariate stable processes is of considerable recent interest in modeling and forecasting independent correlated data. Although several different methods exist in literature for estimating stable law parameters, Bayesian inference for stable processes is virtually unexplored. Except for Buckle\u27s (8) approach for the univariate stable law case, there is no discussion of the Bayesian approach for inference in univariate or multivariate stable processes. By incorporating prior information and performing posterior analysis, the Bayesian approach facilitates simultaneous estimation of the parameters characterizing the stable law, together with the parameters of the correlated process, and enables us to obtain the joint and marginal posterior distributions of all the parameters as well as summary features of these distributions.^ We present Bayesian inference for, (i) univariate stable laws, (ii) univariate autoregressive moving average (ARMA) models with stable innovations, (iii) multivariate stable laws and (iv) vector autoregressive moving average (VARMA) models with stable innovations. In (i) we approximate the posterior density and moments using normal mixtures (West (1993)) and the Laplace approximation (Tierney (1989)) respectively, which are alternatives to Buckle\u27s approach. In (ii), we extend Buckle\u27s approach to time series ARMA models which we compare with the approach using normal mixtures. In (iii), we develop a sampling based Markov chain Monte Carlo approach for multivariate stable laws while in (iv), we extend this approach to vector time series models. In each case, we prove the propriety of the posterior distributions under a non-informative prior specification. We also explicitly incorporate, where necessary, the stationarity and invertibility restrictions on the time series model parameters. To enable sampling from complete conditional distributions that have non-standard forms, we use a Metropolis-Hastings algorithm which, after convergence, gives samples from the required joint posterior. We illustrate our approach through two real data examples for univariate cases and simulated data for multivariate cases.

Bayesian Inference for Time series with Infinite Variance Stable Innovations

This article describes the use of sampling based Bayesian inference for infinite variance stable distributions and for time series with infinite variance stable innovations. For time series, an advantage of the Bayesian approach is that it enables the simultaneous estimation of the parameters characterizing the stable law, together with the parameters of the univariate or multivariate linear ARMA model. Our approach uses a Metropolis- Hastings algorithm to generate samples from the joint posterior distribution of all the parameters and is an extension to univariate and multivariate time series processes of the approach in [Bu] for independent observations. 1. Introduction A random variable X has a stable distribution S(ff; fi; ffi; oe) if there are parameters 0 ! ff 2, \Gamma1 fi 1, oe ? 0 and \Gamma1 ! ffi ! 1 such that its characteristic function has the form ([GK]): E(e itx ) = ae exp(\Gammajoetj ff (1 \Gamma ifisign(t) tan(ßff=2) + iffit) if ff 6= 1 exp(\Gammajoetj(1 + 2if..

Monte Carlo EM estimation for multivariate stable distributions

We describe parameter estimation for the multivariate sub-Gaussian symmetric stable distribution using Monte Carlo EM algorithm. Two augmented vectors are employed in the construction of the posterior joint density of the stable parameters. Gibbs sampling enables the generation of these vectors from their respective conditional posterior distributions and thus facilitates the expectation step of the algorithm.Gibbs sampling Posterior mode Ratio of uniforms Rejection algorithm Sub-Gaussian symmetric stable distribution

Multivariate Survival Analysis with Positive Stable Frailties

this paper, we describe Bayesian modeling of dependent multivariate survival data using positive stable frailty distributions. A flexible baseline hazard formulation using a piecewise exponential model with a correlated prior process is used. The estimation of the stable law parameter together with the parameters of the (conditional) proportional hazards model is facilitated by a modified Gibbs sampling procedure. The methodology is illustrated on kidney infection data. 1. Introductio