Factions in Morris' natural exponential quadratic variance family and a lack of consensus in sample size determination

Morris (1982, 1983) shows that there exists only six distributions with variance as a quadratic function of the mean, and calls the group natural exponential-quadratic variance family (NEQVF). This note discovers that in NEQVF there are actually two factions which we call linear and nonlinear mean exponential families (MEF and NLMEF). A statistical utility of this characterization lies in determining a sample size based on Shannon's information criterion: As far as gathering information with respect to the population mean, there is a consensus formula in NEQVF for sample size determination, but with respect to the natural parameter, there are three opinions -- one for LMEF and a different one for each member of NLMEF.