ON CONSTRUCTING THE NATURAL EXPONENTIAL FAMILIES WITH POLYNOMIAL VARIANCE FUNCTIONS

In this paper,we discuss the regular exponential family(REF),a natural exponentialfamily with minimum dimension and with the mean function m=E_θX taken as a new parameter.Wepoint out the importance of the REF with polynomial variance function(PVF-REF).When m is asingle factor of PVF and the left end of its defined domain is zero,the corresponding REF is givenas the single-side lattice distribution family;other PVF-REFs can be obtained by a limit process.As an illustration,the results for all seven cases of polynomial variance functions of third degreeare given.