We compute a variance lower bound for unbiased estimators in statistical models. The construction of the bound is related to the original Cramér–Rao bound, although it does not require the differentiability of the model. Moreover, we show our efficiency bound to be always greater than the Cramér–Rao bound in smooth models, thus providing a sharper result.
Nous obtenons une minoration pour la variance dʼun estimateur sans biais dans un modèle statistique. La construction de la borne est liée à celle de la borne de Cramér–Rao, mais elle ne nécessite pas dʼhypothèse de différentiabilité sur le modèle. De plus, nous montrons que la borne est toujours supérieure ou égale à la borne de Cramér–Rao dans les modèles différentiables, et fournit ainsi un résultat plus fort.
Accepted:
Published online:
Thibault Espinasse 1; Paul Rochet 1
@article{CRMATH_2012__350_13-14_711_0, author = {Thibault Espinasse and Paul Rochet}, title = {A {Cram\'er{\textendash}Rao} inequality for non-differentiable models}, journal = {Comptes Rendus. Math\'ematique}, pages = {711--715}, publisher = {Elsevier}, volume = {350}, number = {13-14}, year = {2012}, doi = {10.1016/j.crma.2012.08.005}, language = {en}, }
Thibault Espinasse; Paul Rochet. A Cramér–Rao inequality for non-differentiable models. Comptes Rendus. Mathématique, Volume 350 (2012) no. 13-14, pp. 711-715. doi : 10.1016/j.crma.2012.08.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.08.005/
[1] Efficient and Adaptive Estimation for Semiparametric Models, Springer-Verlag, New York, 1998
[2] On topology properties of f-divergences, Studia Sci. Math. Hungar., Volume 2 (1967), pp. 329-339
[3] A Cramér–Rao inequality for non-differentiable models, 2012 | arXiv
[4] A characterization of limiting distributions of regular estimates, Z. Wahr. Verw. Geb., Volume 14 (1969/1970), pp. 323-330
[5] Locally asymptotically normal families of distributions. Certain approximations to families of distributions and their use in the theory of estimation and testing hypotheses, Univ. California Publ. Statist., Volume 3 (1960), pp. 37-98
[6] Information and the accuracy attainable in the estimation of statistical parameters, Bull. Calcutta Math., Volume 37 (1945), pp. 81-91
[7] Asymptotic Statistics, Cambridge University Press, Cambridge, 1998
[8] Semiparametric statistics, Saint-Flour, 1999 (Lectures on Probability Theory and Statistics), Volume 1781 (2002), pp. 331-457
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