Constructing unbiased tests for homogeneity and goodness of fit

Suppose Xij, i=1, 2,..., k, j=1, 2,..., ni, are random samples from independent populations distributed according to an exponential family with parameter [theta]i. Let Yi be the minimal sufficient statistic for population i and assume that the sum of any subset of the Yi, i=1, 2,..., k, is also a one parameter exponential family. The normal and Poisson distributions satisfy such an assumption. The problem is to test H: [theta]1=[theta]2= ... =[theta]k vs K: not H. Unbiased tests are constructed. The construction ca so that the resulting unbiased tests are in a complete class in the continuous case and are admissible in the discrete case. The construction is also appropriate for testing a simple hypothesis concerned with multinomial probabilities against an arbitrary alternative. This latter problem arises in testing goodness of fit.