Order-invariant group reference priors for natural exponential families having a simple quadratic variance function

Reference analysis is one of the most successful general methods to derive noninformative prior distributions. In practice, however, reference priors are often difficult to obtain. Using recently developed theory for conditionally reducible natural exponential families, and the related notion of enriched conjugate prior, we find the group reference prior for the mean- and natural-parameter under the added assumption that the variance function is simple quadratic. Posterior computations are especially straightforward due to the fact that resulting reference distributions belong to the corresponding enriched conjugate family. Moreover, for some parameterizations, these reference priors turn out to be order-invariant. A substantive application of the theory relates to the construction of reference priors for the Bayesian analysis of two-way contingency tables with respect to two alternative parameterizations