Sequential Procedures that Control the Individual Probabilities of Coming to the Various Decisions

Abstract

We consider cases where we have a finite number of decisions and a finite number of possible distributions, and we confine attention to procedures which have zero probability of continuing beyond the $N$th observation, where $N$ is a given positive integer. We find a class $C$ of procedures such that given any procedure $R$, there is a member of $C$, say $R'$, such that the probabilities of coming to the various decisions under the various distributions when using $R'$ are at least as desirable as when using $R$, and such that we are at least as likely to take fewer than $n$ observations under $R'$ as under $R$, for any $n$. Various extensions are indicated.