On the effect of overdispersion on exact conditional tests

Let Y-1,..., Y-n denote independent random variables such that tl has a one -parameter exponential family distribution with canonical parameter theta(j ) = lambda + psi X-j; here X-1,..., X-n are known constants. Consider a tes t of the null hypothesis psi = 0. Under the null hypothesis, A = Sigma Y-j sufficient for lambda and, hence, a test of psi, = 0 may be based on the co nditional distribution of T = Sigma XjYj given A, which is independent of l ambda. In this paper, the effects of overdispersion due to a mixture model on the conditional distribution of T given A are considered.