Let f be a probability density on the real line, let n be any positive integer, and assume the condition (R) that logf is locally integrable with respect to Lebesgue measure. Then either logf 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≥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 Schwartz (1947).
Publié le : 2000-12-14
Classification:
characterization,
complete sufficient statistics,
equivariance,
exponential family, independence,
infinitely divisible distribution,
mean periodic functions,
normal distribution,
order statistics,
transformation model
@article{1081194163,
author = {Mattner, Lutz},
title = {Minimal sufficient statistics in location-scale parameter models},
journal = {Bernoulli},
volume = {6},
number = {6},
year = {2000},
pages = { 1121-1134},
language = {en},
url = {http://dml.mathdoc.fr/item/1081194163}
}
Mattner, Lutz. Minimal sufficient statistics in location-scale parameter models. Bernoulli, Tome 6 (2000) no. 6, pp. 1121-1134. http://gdmltest.u-ga.fr/item/1081194163/