Construction of minimax tests for bounded families of probability densities (Q1209846)

Let \(f\) denote the Radon-Nikodym density of a probability measure with respect to a \(\sigma\)-finite measure \(\mu\) and let \(\underline f, {\underset {=} f}, \underline g, {\underset {=} g}\) be nonnegative, measurable functions satisfying \[ \oint \underline f d \mu \leq 1 \leq \oint {\underset {=} f} d \mu \quad \text{and} \quad \oint \underline g d \mu \leq 1 \leq \oint {\underset {=} g} d \mu. \] The compound test problem under consideration, \[ H_ 0 : \underline f \leq f \leq {\underset {=} f} \text{ vs. } H_ 1 : \underline g \leq g \leq {\underset {=} g}, \] is dealt with by using the Neyman-Pearson theory for single hypotheses and the risk-function method. Least favourable pairs of distributions are constructed as well as a family of minimax tests.