Sufficient Statistics and Exponential Families

Using a locally Lipschitz function T of $n > 1$ variables one can reduce data consisting of a sample of size n to one real number. If we are given a family of probability measures on the real line which are equivalent to Lebesgue measure then T yields a sufficient data reduction only if the given family is exponential. This result is compared with the results of Brown (1964) and Denny (1970).