UNE INTRODUCTION À LA THÉORIE GÉNÉRALE DE L’APPROXIMATION
QUADRATIQUE D’UNE APPLICATION LINÉAIRE
Abstract: The linear statistical inference constitutes one of the more usual frameworks for
quadratic approximation of linear mappings. Generally, the set of observations is a Euclidean
space and the selected risk function is the trace of expectation of a quadratic loss
function. In fact, most common notions which one can manipulate there (Gauss-Markov’s
estimation for example) are totally independent of all but linear structures on the space of
the observations.
We would like to introduce here the general problem of the quadratic approximation of a
linear mapping of a finite-dimensional vector space into another finite-dimensional vector
space including that of the admissibility of solutions, without any other hypothesis other
than the spaces and each being placed in separate duality with another vector space.
We show how the positivity of the considered operators and the provision of a Euclidean
structure on the image space is sometimes necessary to assure the non-emptiness of the
set of solutions. We would like to end by giving, within this general framework, a proof of an
extension of a fundamental L. R. LaMotte [5] theorem based on the Hahn-Banach
theorem.
2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;
Key words and phrases: -