Data based loss estimation of the mean of a spherical distribution with a residual vector

Abstract In the canonical setting of the general linear model, we are concerned with estimating the loss of a point estimator when sampling from a spherically symmetric distribution. More precisely, from an observable (X, U) in %${\mathbb {R}}^p \times {\mathbb {R}}^k%$ having a density of the form...
Ausführliche Beschreibung

Abstract In the canonical setting of the general linear model, we are concerned with estimating the loss of a point estimator when sampling from a spherically symmetric distribution. More precisely, from an observable (X, U) in %${\mathbb {R}}^p \times {\mathbb {R}}^k%$ having a density of the form %$1 / \sigma ^{p+k} \, f \! \left( \big ( \Vert {\textbf{x}}- \varvec{\theta }\Vert ^2 + \Vert {\textbf{u}}\Vert ^2 / \sigma ^2 \big ) \right) %$ where %$\varvec{\theta }%$ and %$\sigma %$ are both unknown, we consider general estimators %$ \varphi (X,\Vert U\Vert ^2) %$ of %$\varvec{\theta }%$ under two losses: the usual quadratic loss %$\Vert \varphi (X,\Vert U\Vert ^2) - \varvec{\theta }\Vert ^2%$ and the data-based loss %$\Vert \varphi (X,\Vert U\Vert ^2) - \varvec{\theta }\Vert ^2 / \Vert U\Vert ^2%$. Then, for each loss, we compare, through a squared error risk, their unbiased loss estimator %$\delta _0(X,\Vert U\Vert ^2)%$ with a general alternative loss estimator %$\delta (X,\Vert U\Vert ^2)%$. Thanks to the new Stein type identity in Fourdrinier and Strawderman (Metrika 78(4):461–484, 2015), we provide an unbiased estimator of the risk difference between %$\delta (X,\Vert U\Vert ^2)%$ and %$\delta _0(X,\Vert U\Vert ^2)%$, which gives rise to a sufficient domination condition of %$\delta (X,\Vert U\Vert ^2)%$ over %$\delta _0(X,\Vert U\Vert ^2)%$. Minimax estimators of Baranchik form illustrate the theory. It is found that the distributional assumptions and dimensional conditions on the residual vector U are weaker when the databased loss is used. Ausführliche Beschreibung

Autor*in:	Canu, Stéphane [verfasserIn] Fourdrinier, Dominique

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2023

Schlagwörter:	Data based loss Loss estimation Spherically symmetric distributions Stein type identity

Anmerkung:

© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Übergeordnetes Werk:	Enthalten in: Metrika - Berlin : Springer, 1958, 86(2023), 8 vom: 06. Feb., Seite 851-878
Übergeordnetes Werk:	volume:86 ; year:2023 ; number:8 ; day:06 ; month:02 ; pages:851-878

Links:	Volltext

DOI / URN:	10.1007/s00184-023-00895-4

Katalog-ID:	SPR053094832