Jeffreys versus Shtarkov distributions associated with some natural exponential families

Jeffreys and Shtarkov distributions play an important role in universal coding and minimum description length (MDL) inference, two central areas within the field of information theory. It was recently discovered that in some situations Shtarkov distributions exist while Jeffreys distributions do not. To demonstrate some of these situations we consider in this note the class of natural exponential families (NEF’s) and present a general result which enables us to construct numerous classes of infinitely divisible NEF’s for which Shtarkov distributions exist and Jeffreys distributions do not. The method used to obtain our general results is based on the variance functions of such NEF’s. We first present two classes of parametric NEF’s demonstrating our general results and then generalize them to obtain numerous multiparameter classes of the same type.