A solution to multivariate state-space modeling, forecasting, and smoothing is discussed. We allow for the possibilities of nonnormal errors and nonlinear functionals in the state equation, the observational equation, or both. An adaptive Monte Carlo integration technique known as the Gibbs sampler is proposed as a mechanism for implementing a conceptually and computationally simple solution in such a framework. The methodology is a general strategy for obtaining marginal posterior densities of coefficients in the model or of any of the unknown elements of the state space. Missing data problems (including the k-step ahead prediction problem) also are easily incorporated into this framework. We illustrate the broad applicability of our approach with two examples: a problem involving nonnormal error distributions in a linear model setting and a one-step ahead prediction problem in a situation where both the state and observational equations are nonlinear and involve unknown parameters.
Bibliographical noteFunding Information:
• Bradley P. Carlin is Assistant Professor, Division of Biostatistics, School of Public Health, University of Minnesota, Minneapolis, MN 55455. Nicholas G. Polson is Assistant Professor, Graduate School of Business, University of Chicago, Chicago, IL 60637. David S. Stoffer is Associate Professor, Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, PA 15260. The work of Carlin was supported in part by National Science Foundation Grant DMS 88-05676; the work of Stoffer was supported in part by National Science Foundation Grant DMS 90-00522 and by a grant from the Centers for Disease Control through a cooperative agreement with the Association of Schools of Public Health. This research was done while all three authors were visitingthe Department of Statistics at Carnegie Mellon University.
- Gibbs sampler
- Kalman filter