This paper is concerned with testing the hypothesis that the parameter in a multivariate exponential distribution lies in a linear subspace of the natural parameter space. Our main result characterizes a complete class of tests which is independent of the particular exponential distribution. This class is, in fact, complete relative to the stronger ordering among tests which compares conditional power, given a certain statistic, pointwise. The conclusion holds without any restriction on the exponential distribution. Many of the tests are admissible, but examples show that although the class is essentially the smallest class complete relative to all exponential distributions, it is not in general minimally complete. Some special cases where the class is minimally complete are discussed.
"Tests of Composite Hypotheses for the Multivariate Exponential Family." Ann. Math. Statist. 38 (3) 681 - 697, June, 1967. https://doi.org/10.1214/aoms/1177698862