A note on overdispersed exponential families

The issue of creating overdispersion in a given one-parameter one-dimensional exponential family, by extending it to a two-parameter exponential family with the same support, is considered. An easily verifiable sufficient condition for this is derived. It is shown that a large class of families satisfy this condition and that this class includes Efron's (1986) and Lindsay's (1986) family as special cases. This class is also closely related to Jorgensen's (1987) exponential dispersion models. Uniformly most powerful unbiased tests for testing overdispersion are exhibited and it is shown that in this context Cox's overdispersion test is a special case. A graphical display is developed to select a family of the general class which should be used with a given set of data to overdisperse a given target one-parameter family. Two real illustrations are given.