Sufficiency,generalized likelihood function and exponential family

The problem of characterizing an exponential family by sufficiency of certain statistics is considered. In distinction to most of the papers on this subject we do not want to characterize an exponential family of order less than or equal to k by the existence of a-dimensional sufficient statis tics in.GepenoenX' oi inc sample size. Since such a characterization is only valid under regularity assumptions, which is shown in the paper, we consider a stronger property of an exponential family which turns out to be a characteristic one. At this the concept of generalized likelihood functions will play an important role.