Asymptotic Optimality of Certain Multi - Alternativesequential Tests

It is known that certain combinations of one-sided sequential probability ratio tests are asymptotically optimal (relative to the expected sample size) for problems involving a nite number of possible distributions when probabilities of errors tend to zero and observations are independent and identically distributed according to one of the underlying distributions. The objective of this paper is to show that two speciic constructions of sequential tests asymptotically minimize not only the expected time of observation but also any positive moment of the stopping time distribution under fairly general conditions for a nite number of simple hypotheses. This result appears to be true for general statistical models which include correlated and inhomogeneous processes observed either in discrete or continuous time. For statistical problems with nuisance parameters, we consider invariant sequential tests and show that the same result is valid for this case. Finally, we apply general results to the solution of several particular problems such as multi-sample slippage problem for correlated Gaussian processes and for statistical models with nuisance parameters.