Abstract
The principal object of the investigation is a family of scalar products defined on a Hilbert space ℋ. For this general case of the theory of linear statistical inference, we obtain a counterpart of the Neyman-Fisher factorization theorem, and we introduce and study counterparts of the Fisher information and of the information matrix that preserve all the important properties of their classical prototypes. By additionally introducing the concept of observation interpreted as a linear functional on ℋ that satisfies a certain additional condition, we can construct a natural counterpart of the maximum likelihood method.
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 53, pp. 64–100, 1975.
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Gerlein, O.V., Kagan, A.M. Hilbert space methods in classical problems of mathematical statistics. J Math Sci 12, 184–213 (1979). https://doi.org/10.1007/BF01262718
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DOI: https://doi.org/10.1007/BF01262718