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: -;

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