Minimal Suucient 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 suucient. It follows, subject to (R) and n 3, that a complete suucient statistic exists in the normal case only. Also, for f with (R) innnitely divisible but not normal, the order statistic is always minimal suucient 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 K.