The Γ-minimax estimator under squared error loss for the unknown parameter of a one-parameter exponential family with an unbiased sufficient statistic having a variance which is quadratic in the parameter is explicitly determined for a class Γ of priors consisting of all distributions whose first two moments are within some given bounds. This generalizes the choice of Γ in Jackson et al. (1970) as well as the unrestricted case. It is shown that the underlying statistical game is always strictly determined and that there exists a Γ-minimax estimator which is a linear function of the unbiased sufficient statistic. If the bounds for both prior moments are effective then there exists a least favourable prior in Γ which is a member of the Pearsonian family.
CONTENTS1. Introduction and summary....................52. A class of exponential families..............63. The estimation problem......................124. Solution of the statistical games.........175. Some special cases............................326. Concluding remark.............................33References............................................35
1980 Mathematics Subject Classification: (1985 Revision): Primary 62C99; Secondary 62F10.
@book{bwmeta1.element.zamlynska-f53b6f38-8c1f-4764-82f9-f71400ad28a0,
author = {Lanxiang Chen and Heike Hofmann and J\"urgen Eichenauer-Herrmann and J\"urgen Kindler},
title = {Gamma-minimax estimators in the exponential family},
series = {GDML\_Books},
publisher = {Instytut Matematyczny Polskiej Akademi Nauk},
address = {Warszawa},
year = {1991},
zbl = {0721.62014},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.zamlynska-f53b6f38-8c1f-4764-82f9-f71400ad28a0}
}
Lanxiang Chen; Heike Hofmann; Jürgen Eichenauer-Herrmann; Jürgen Kindler. Gamma-minimax estimators in the exponential family. GDML_Books (1991), http://gdmltest.u-ga.fr/item/bwmeta1.element.zamlynska-f53b6f38-8c1f-4764-82f9-f71400ad28a0/