We investigate a logarithmic-link generalized linear model, whose underlying sam- pling density is in an exponential fainily distribution with power variance function. The multiple-strata case is studied with stratum-dependent intercepts and a common slope. We prove that there exists a conjugate prior density on the intercept parameter, and the conjugate analysis is discussed. An estimation procedure is given, which incIudes the optimal estimating function of the parameters other than the intercept, and an empirical Baysian estimation of the hyper-parameters of the prior density. As an example, rainfall data for Queensland, Australia, is analyzed.
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