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PARAMETRIC STATISTICAL UNCERTAINTY RELATIONS AND

PARAMETRIC STATISTICAL FUNDAMENTAL EQUATIONS

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T. MATSUNAWA

*The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu,*

Minato-ku, Tokyo 106-8569, Japan
(Received November 21, 1996; revised October 15, 1997)

**Abstract.**
Multivariate parametric statistical
uncertainty relations are
proved to specify multivariate
basic parametric statistical models. The relations are
expressed by inequalities. They generally show that we cannot
exactly determine simultaneously both a function of
observation objects and a parametric statistical model in a
compound parametric statistical system composed of
observations and a model. As special cases of the relations,
statistical fundamental equations are presented which are
obtained as the conditions of attainment of the equality sign
in the relations. Making use of the result, a generalized
multivariate exponential family is derived as a family of
minimum uncertainty distributions. In the final section,
several multivariate distributions are derived as basic
multivariate parametric statistical models.

*Key words and phrases*:
Parametric statistical
uncertainty relations, parametric statistical fundamental
equations, specification of basic parametric statistical
models, specific intensity of model performance, primary
parameter, secondary parameter, minimum statistical
uncertainty distributions.

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