**Title**: A Group-Theoretical Approach Towards Envelopes

**Abstract**:

Dennis Cook’s envelope model represents a major contribution to the statistical theory of prediction. In its x-reduction variant it corresponds to the partial least squares model, and it is a reduction of the multivariate regression model. In my talk I will address the following question: Can such a reduction be motivated intuitively? My suggestion is to use symmetry considerations, more concretely, use group theory. Group actions on the parameter space are defined in general, and orbits of such groups are introduced. As a general principle put forward, it is claimed that every model reduction should be to an orbit or to a set of orbits of the group. This principle is motivated, and shown to give natural model reductions in several simple cases. In the multivariate regression case with a reasonable group defined on the parameter space, it leads to an envelope model. Next a generalized Pitman theorem is formulated, and used as a motivation to introduce a reasonable prior in the envelope model. The corresponding Bayes estimator is briefly described.