A Unified Approach to Prior and Loss Robustness

ABSTRACT

We introduce a new statistical framework in order to study Bayesian loss robustness under classes of priors distributions, thus unifying both concepts of robustness. We propose measures that capture variation with respect to both prior selection and selection of loss function and explore general properties of these measures. We illustrate the approach for the continuous exponential family. Robustness in this context is studied first with respect to prior selection where we consider several classes of priors for the parameter of interest, including unimodal and symmetric and unimodal with positive support. After prior variation has been measured we investigate robustness to loss function, using Hellinger and Linex (Linear Exponential) classes of loss functions. The methods are applied to standard examples.

Keywords:

• Continuous exponential family
• Hellinger loss
• Linex loss
• Loss robustness
• Posterior expected loss

Mathematics Subject Classification:

• 62F99

Acknowledgments

The author is grateful to Professors Dipak K. Dey and Farroll T. Wright for helpful suggestions regarding an earlier version of the manuscript.