Minimal sufficient statistics in location-scale parameter models

Let f be a probability density on the real line, let n be any positive integer, and assume the condition (R) that log f is locally integrable with respect to Lebesgue measure. Then either log f is almost everywhere equal to a polynomial of degree less than n, or the order statistic of n independent and identically distributed observations from the location-scale parameter model generated by f is minimal sufficient. It follows, subject to (R) and n greater than or equal to 3, that a complete sufficient statistic exists in the normal case only. Also, for f with (R) infinitely divisible but not normal, the order statistic is always minimal sufficient for the corresponding location-scale parameter model. The proof of the main result uses a theorem on the harmonic analysis of translation- and dilation-invariant function spaces, attributable to Leland (1968) and to Schwartz (1947).