A note on insufficiency and the preservation of Fisher information. (English) Zbl 1346.62006

Pollard, David

Banerjee, M. (ed.) et al., From probability to statistics and back: high-dimensional models and processes. A Festschrift in honor of Jon A. Wellner. Including papers from the conference, Seattle, WA, USA, July 28–31, 2010. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 978-0-940600-83-6). Institute of Mathematical Statistics Collections 9, 266-275 (2013).

Summary: A. Kagan and L. A. Shepp [“A sufficiency paradox: an insufficient statistic preserving the Fisher information”, Am. Stat. 59, No. 1, 54–56 (2005), http://www.jstor.org/stable/27643616] presented an elegant example of a mixture model for which an insufficient statistic preserves Fisher information. This note uses the regularity property of differentiability in quadratic mean to provide another explanation for the phenomenon they observed. Some connections with Le Cam’s theory for convergence of experiments are noted.
For the entire collection see [Zbl 1319.62002].

MSC:

62B10

Statistical aspects of information-theoretic topics

Keywords:

Fisher information; sufficiency; Hellinger differentiability of probability models; differentiability in quadratic mean; score function; Le Cam’s distance between statistical models