Inference Robustness in Multivariate Models with a Scale Parameter

We formulate a general representation of points z E Rn - {O} in terms of pairs (y, r), where r > 0, y lies in some space y, and z = ry. In addition, we impose that the representation is unique. An example of such a representation is polar coordinates. As an immediate consequence, we can represent random variables Z that take values in Rn - {O} as Z = RY, where R is a positive random variable and Y takes values in y. By fixing the distribution of either R or Y, while imposing independence between them, we generate classes of distributions on Rn. Many families of multivariate distributions, like e.g. spherical, lq-spherical, v-spherical and anisotropic, can be interpreted in this unifying framework. Some classical inference procedures can be shown to be completely robust in these classes of multivariate distributions. These findings are used in the practically relevant contexts of location-scale and pure scale models. Finally, we present a robust Bayesian analysis for the same models and indicate the links between classical and Bayesian results. In particular, for the regression model with Li.d. errors up to a scale: a formal characterization is provided for both classical and Bayesian robustness results concerning inference on the regression parameters.