Some Applications to Statistical Decision Theory

Abstract

The mathematical models to be introduced in the sequel concern certain statistical decision problems. A statistician has carried out a random experiment and has to make a decision basing on the information gained by the experiment. The informations are controlled by a probability distribution which is not known to the statistician. Nevertheless, there is a collection of distributions under consideration, which come into question to control the random experiment. The space of possible results of the random experiment is assumed to be a measurable space (H,H), which is called sample space. The collection of possible sample distributions is assumed to be given by a stochastic kernel

$$ P:\Gamma \times H \to [0,1]$$

where (r,G) is a measurable space, and is called parameter space. Further, we presuppose that the collection

$$\omega: = \{P(\gamma ,.)|\gamma \in \Gamma \} $$

of possible sample distributions is dominated by some probability measure µ on H, and that there exists some measurable function

$$ f:IH \times \Gamma \to {\mathbb{R}_{ + }} $$