A CLASS OF ASYMPTOTICALLY OPTIMAL SEQUENTIAL-TESTS FOR COMPOSITE HYPOTHESES

Suppose that X has density {f(x, theta)=exp(theta x-psi(theta)} (with respect to some measure nu), where theta is an element of (<(theta)und er bar>, <(theta)over bar>, - infinity < less than or equal to <(theta )under bar> < <(theta)over bar> less than or equal to + infinity. Cons ider the problem of testing the hypothesis theta less than or equal to theta(0) against theta greater than or equal to theta(1) for some giv en theta(0) and theta(1) (<(theta)under bar> < theta(0) < theta(1) < < (theta)over bar>). A class of truncated sequential tests is introduced with type I and type II error probabilities not exceeding alpha and b eta, respectively. The expected sample sizes of these tests are shown to be asymptotically minimal for all theta as alpha + beta down arrow 0.