Suuciency, Exponential Families, and Algebraically Independent Numbers

We construct a continuous, strictly increasing, and bounded function T which maps Lebesgue almost the entire real line onto an algebraically independent set of real numbers. It follows that P n i=1 T (X i) is a uniformly continuous one-dimensional suucient statistic for every IID{model P n = fP n : P 2 Pg such that each P 2 P has a Lebesgue density. It also follows that the one{parameter exponential family P with Lebesgue densities proportional to, say, exp(?x 2 + #T (x)) is such that for the corresponding P n the order statistic is complete. Hence, in spite of exponentiality, P n does not admit a suucient reduction beyond the trivial one. Under suitable stronger regularity conditions, such as continuous diierentiability of T , none of the above is possible.