When Gaussian errors are inappropriate in a multivariate linear regression setting, it is often assumed that the errors are iid from a distribution that is a scale mixture of multivariate normals. Combining this robust regression model with a default prior on the unknown parameters results in a highly intractable posterior density. Fortunately, there is a simple data augmentation (DA) algorithm and a corresponding Haar PX-DA algorithm that can be used to explore this posterior. This paper provides conditions (on the mixing density) for geometric ergodicity of the Markov chains underlying these Markov chain Monte Carlo algorithms. Letting d denote the dimension of the response, the main result shows that the DA and Haar PX-DA Markov chains are geometrically ergodic whenever the mixing density is generalized inverse Gaussian, log-normal, inverted Gamma (with shape parameter larger than d/2) or Fréchet (with shape parameter larger than d/2). The results also apply to certain subsets of the Gamma, F and Weibull families.
Bibliographical noteFunding Information:
The authors thank two anonymous referees for helpful comments and suggestions. The first author was supported by NSF grants DMS-11-06395 and DMS-15-11945, and the third by NSF grant DMS-11-06084 and DMS-15-11945.
© 2018 Board of the Foundation of the Scandinavian Journal of Statistics
- Haar PX-DA algorithm
- data augmentation algorithm
- drift condition
- geometric ergodicity
- heavy-tailed distribution
- minorization condition
- scale mixture