Statistical Decision Rules and Optimal Inference

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Statistical Decision Rules and Optimal Inference

Statistical Decision Rules and Optimal Inference
Nikolai Nikolaevich Cencov,
1982 | ISBN-10: 0821845020 | 499 pages | Djvu | 7 MB

The general concepts of a statistical decision and a statistical decision rule are basic for all of modem statistical theory. According to Wald, every
particular statistical problem is a problem of decision-making: the statistician, having processed certain observational material, must draw conclusions as to the observed phenomenon. Since the outcome of each observation is random., one cannot usually expect these conclusions to be absolutely accurate. It is a job for the theory to ascertain the minim al unavoidable uncertainty of the conclusions in the problem and to indicate an optimal decision rule.
In classical problems of mathematical statistics, one is required to determine the (unknown) probability distribution of the outcomes on the basis
of independent observations and certain additional information. When the number of observations used in such cases increases, one can establish
various simple and quite general asymptotic relationships.
In any theory, a general law should be amenable to equivalent formulations; that is to say, the statement of the law should not vary when a situation
is replaced by another one, equivalent to the former (within the framework of that theory); otherwise it would not be a general law. In classical geometry, such "changes of situation" form a group. In mathematical statistics the description of the set of equivalent situations is more complicated.
The system of all statistical decision rules for all conceivable statistical problems, together with the natural operation of composition, forms an
algebraic category. This category generates a uniform geometry of f amili es of probability laws, in which the "figures" are the f amili es and the "motions" are the decision rules. Two families are "congruent" if and only if they possess equivalent statistical properties...

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Tags: Optimal, Inference, Decision, Statistical
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