Minimal Sufficient Statistics in Location-Scale Parameter Models

Abstract

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 su cient. It follows, subject to (R) and n 3, that a complete su cient statistic exists in the normal case only. Also, for f with (R) in nitely divisible but not normal, the order statistic is always minimal su cient 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.O. Leland (1968) and L. Schwartz (1947).

Minimal sufficiency of order statistics in convex models

“…Since this is not always easy, it appears to be worthwhile to supply theorems yielding minimal sufficient σ-algebras for statistically interesting classes of models. Restricting attention to models for independent and identically distributed observations, we note that the case of exponential families is well understood (see, for example, Theorem 1.6.9 in Pfanzagl [12] and, concerning a possible misinterpretation, Theorem 2.3 of Mattner [10]) and that the case of location-scale parameter models on the real line has been treated in Mattner [11].…”

“…To prove (2) ⇒ (3), we may use the following beautiful example, indicated on page 18 of Torgersen [15] and explained in more detail in Remarks 1.2 and 1.3 of Mattner [11]: Let f be a probability density with respect to Lebesgue measure λ λ λ on R, such that f = f 1 f 2 with f 1 a normal density and f 2 periodic, and let P be the corresponding location parameter model, P = {f (· − ϑ)λ λ λ : ϑ ∈ R}. Then, for every n ≥ 2, the order statistic is not minimal sufficient for P n but, except for very special and explicitly known f 2 , the model P is not an exponential family.…”