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Complete classes of tests for some nonparametric hypotheses. (English) Zbl 0820.62005

Let \({\mathcal X}\) be a sample space with a \(\sigma\)-field \({\mathcal F}\) and let \(\mu\) be a \(\sigma\)-finite measure on \({\mathcal F}\). We assume that \(\Omega\) is a family of probability measures which are absolutely continuous with respect to \(\mu\). We bring out some direct relationships between the problem of testing a null hypotheses \(H_ 0 : P \in \Omega_ 0 \subseteq \Omega\) versus an alternative \(H_ A : P \in \Omega_ A \subseteq \Omega\) and the theory of weak topologies in Banach spaces \(L^ 1 ({\mathcal X}, {\mathcal F}, \mu)\) and \(L^ \infty ({\mathcal X}, {\mathcal F}, \mu)\).
In Section 2 we prove an essentially complete class theorem for weakly compact topologically separated subsets of \(L^ 1 ({\mathcal X}, {\mathcal F}, \mu)\) space. We apply this result to testing some approximate hypotheses which may give some insights in the problem of robust tests. We show that in the case considered, the approach based on Dirichlet processes is not applicable. We also examine the above result in the light of results for Choquet capacities.
Finally, in Section 3, we present a simple proof that for testing arbitrary hypotheses, the set of all \(\text{weak}^*\) limits of convergent sequences of Bayes tests constitutes an essentially complete class, no matter what \(\sigma\)-field \({\mathcal F}\) is.

MSC:

62C07 Complete class results in statistical decision theory
62G10 Nonparametric hypothesis testing
62C15 Admissibility in statistical decision theory
46N30 Applications of functional analysis in probability theory and statistics
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