TY - JOUR

T1 - Robust minimax Stein estimation under invariant data-based loss for spherically and elliptically symmetric distributions

AU - Fourdrinier, Dominique

AU - Strawderman, William

N1 - Funding Information:
This work was partially supported by a Grant from the Simons Foundation (#209035 to William Strawderman).
Publisher Copyright:
© 2014, Springer-Verlag Berlin Heidelberg.

PY - 2015/5/1

Y1 - 2015/5/1

N2 - From an observable (X, U) in (Formula Presented.), we consider estimation of an unknown location parameter (Formula Presented.) under two distributional settings: the density of (X, U) is spherically symmetric with an unknown scale parameter σ and is ellipically symmetric with an unknown covariance matrix Σ. Evaluation of estimators of θ is made under the classical invariant losses (Formula Presented.) as well as two respective data based losses (Formula Presented.) where (Formula Presented.). We provide new Stein and Stein–Haff identities that allow analysis of risk for these two new losses, including a new identity that gives rise to unbiased estimates of risk (up to a multiple of (Formula Presented.) in the spherical case for a larger class of estimators than in Fourdrinier et al. (J Multivar Anal 85:24–39, 2003). Minimax estimators of Baranchik form illustrate the theory. It is found that the range of shrinkage of these estimators is slightly larger for the data based losses compared to the usual invariant losses. It is also found that X is minimax with finite risk with respect to the data-based losses for many distributions for which its risk is infinite when calculated under the classical invariant losses. In these cases, including the multivariate t and, in particular, the multivariate Cauchy, we find improved shrinkage estimators as well.

AB - From an observable (X, U) in (Formula Presented.), we consider estimation of an unknown location parameter (Formula Presented.) under two distributional settings: the density of (X, U) is spherically symmetric with an unknown scale parameter σ and is ellipically symmetric with an unknown covariance matrix Σ. Evaluation of estimators of θ is made under the classical invariant losses (Formula Presented.) as well as two respective data based losses (Formula Presented.) where (Formula Presented.). We provide new Stein and Stein–Haff identities that allow analysis of risk for these two new losses, including a new identity that gives rise to unbiased estimates of risk (up to a multiple of (Formula Presented.) in the spherical case for a larger class of estimators than in Fourdrinier et al. (J Multivar Anal 85:24–39, 2003). Minimax estimators of Baranchik form illustrate the theory. It is found that the range of shrinkage of these estimators is slightly larger for the data based losses compared to the usual invariant losses. It is also found that X is minimax with finite risk with respect to the data-based losses for many distributions for which its risk is infinite when calculated under the classical invariant losses. In these cases, including the multivariate t and, in particular, the multivariate Cauchy, we find improved shrinkage estimators as well.

KW - Data-based losses

KW - Elliptically symmetric distributions

KW - Location parameter

KW - Minimaxity

KW - Spherically symmetric distributions

KW - Stein type identity

KW - Stein–Haff type identity

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U2 - 10.1007/s00184-014-0512-x

DO - 10.1007/s00184-014-0512-x

M3 - Article

AN - SCOPUS:84939877323

VL - 78

SP - 461

EP - 484

JO - Metrika

JF - Metrika

SN - 0026-1335

IS - 4

ER -