- Volume 6, Number 6 (2000), 1121-1134.
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 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).
Bernoulli, Volume 6, Number 6 (2000), 1121-1134.
First available in Project Euclid: 5 April 2004
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characterization complete sufficient statistics equivariance exponential family, independence infinitely divisible distribution mean periodic functions normal distribution order statistics transformation model
Mattner, Lutz. Minimal sufficient statistics in location-scale parameter models. Bernoulli 6 (2000), no. 6, 1121--1134. https://projecteuclid.org/euclid.bj/1081194163