Summary: The problem of sequential testing of multiple hypotheses is considered, and two candidate sequential test procedures are studied. Both tests are multihypothesis versions of the binary sequential probability ratio test (SPRT), and are referred to as MSPRTs. The first test is motivated by Bayesian optimality arguments, while the second corresponds to a generalized likelihood ratio test. It is shown that both MSPRTs are asymptotically optimal relative not only to the expected sample size but also to any positive moment of the stopping time distribution, when the error probabilities or, more generally, risks associated with incorrect decisions are small. The results are first derived for the discrete-time case of independent and identically distributed (i.i.d.) observations and simple hypotheses. They are then extended to general, possibly continuous-time, statistical models that may include correlated and nonhomogeneous observation processes. It also demonstrated that the results can be extended to hypothesis testing problems with nuisance parameters, where the composite hypotheses, due to nuisance parameters, can be reduced to simple ones by using the principle of invariance. These results provide a complete generalization of the results given by V. V. Veeravalli and C. W. Baum [see IEEE Trans. Inf. Theory 41, No. 6, Pt. 2, 1994–1997 (1995; Zbl 0844.62069)], where it was shown that the quasi-Bayesian MSPRT is asymptotically efficient with respect to the expected sample size for i.i.d. observations.
For Part II, see IEEE Trans. Inf. Theory 46, No. 4, 1366–1383 (2000; Zbl 1059.62581).