Parametric Statistical Uncertainty Relations and Parametric Statistical Fundamental Equations

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.