Les aspects fondamentaux de l'admissibilité en approximation quadratique d'applications linéaires

Abstract

The quadratic approximation of a linear mapping under affine constraints is a central problem in linear statistical inference. It is based on two fundamental orderings on the set of quadratic forms defined on a finite dimensional vector space E: one is Loewner ordering, and the other based on the trace of linear mappings of E into E. The direction of the affine subspace of constraints is a tensor product of two vector spaces. The search for minimum solutions is now replaced by investigation of minimal solutions, called optimal or admissible according to the ordering. A functional approach shows that we can obtain many identical results which often only depend on the linear structure of the considered vector spaces. We prove thereby two results: The first is a significant generalization of a theorem due to A. Olsen, J. Seely, and D. Birkes [11] on the completeness of the set of optimal or admissible solutions, and the second is theorem containing some necessary conditions of admissibility. This theorem is a consequence of an extension of an important theorem due to L. R. LaMotte [9], proof of which has already been given in this general framework ([18], [19]).