SummaryConsistency and asymptotic normality of the maximum likelihood estimator of β in the loglinear model E(yi) = eα+βXi, where yiare independent Poisson observations, 1iaan, are proved under conditions which are near necessary and sufficient. The asymptotic distribution of the deviance test for β=β0is shown to be chi‐squared with 1 degree of freedom under the same conditions, and a second order correction to the deviance is derived. The exponential model for censored survival data is also treated by the same m