UNBIASED TESTING IN EXPONENTIAL FAMILY REGRESSION

Let (X(ij), z(i)), i = 1, 2, ..., k, j = 1, 2,..., n(i), be independen t observations such that z(i) is a fixed r x 1 vector [r can be 0 (no z's observed) or 1, 2, ..., k - 1], and X(ij) is distributed according to a one-parameter exponential family which is log concave with natur al parameter theta(i). We test the hypothesis that theta = Z beta, whe re theta = (theta(1), ..., theta(k))', Z is the matrix whose ith row i s z'(i) and beta = (beta(1), ..., beta r)' is a vector of parameters. We focus on r = 2 and z'(i) = (1,z(i)), i = 1, 2, ..., k,z(i) < z(i+1) . The null hypothesis on hand is thus of the form theta(i) = beta(1) beta(2)z(i). In such a case the model under the null hypothesis becom es logistic regression in the binomial case, Poisson regression in the Poisson case and linear regression in the normal case. We consider mo stly the one-sided alternative that the second-order differences of th e natural parameters are nonnegative. Such testing problems test goodn ess of fit vs. alternatives in which the natural parameters behave in a convex way. We find classes of tests that are unbiased and that lie in a complete class. We also note that every admissible test of consta nt size is unbiased. In some discrete situations we find the minimal c omplete class of unbiased admissible tests. Generalizations and exampl es are given.