# Codes and Expansions (CodEx) Seminar

## Doug Cochran (Arizona State University)

Geometry of Invariants for Generalized Coherence Tests

This paper considers maximal invariant statistics for testing the null hypothesis that M sets each containing K jointly normal complex N-vectors are statistically mutually independent against its alternative. Invariance of this testing problem is described first in terms of the data space, a collection of M non-singular NxK data matrices, or equivalently a collection of subspaces of the data space (i.e., points on a complex Grassmannian manifold). Invariance of the testing problem is also examined in terms of a parameter space consisting of a collection of positive definite matrices. The full invariance group is derived from both perspectives. The probability distribution of the maximal invariant under action of a particular subgroup of the full invariance group is obtained under both the null and alternative hypotheses. In the parameter, the maximal invariants under the action of this subgroup form a compact space on which proper non-informative prior distributions (e.g., to enable Bayesian analysis) can be constructed.

This is joint work with Stephen Howard and Songsri Sirianunpiboon.