Lindley Type Estimators with the Known Norm

• Baek, Hoh-Yoo (School of Mathematical Science, Wonkwang University)
• Published : 2000.04.30

Abstract

Consider the problem of estimating a $p{\times}1$ mean vector ${\underline{\theta}}(p{\geq}4)$ under the quadratic loss, based on a sample ${\underline{x}_{1}},\;{\cdots}{\underline{x}_{n}}$. We find an optimal decision rule within the class of Lindley type decision rules which shrink the usual one toward the mean of observations when the underlying distribution is that of a variance mixture of normals and when the norm ${\parallel}\;{\underline{\theta}}\;-\;{\bar{\theta}}{\underline{1}}\;{\parallel}$ is known, where ${\bar{\theta}}=(1/p){\sum_{i=1}^p}{\theta}_i$ and $\underline{1}$ is the column vector of ones.

References

1. Annals of Statistics v.6 The geometry of exponential families Efron, B.
2. Biometrika v.64 Conditional inference about a normal mean with known coefficient of variation Hinkley, D. V.
3. Communication in Statistics-Theory and Methods v.22 no.10 James-Stein estimation with constraints on the norm Marchand, E.;Giri, N. C.
4. Journal of Multivariate Analysis v.4 Minimax estimation of location parameters for certain spherically symmetric distributions Strawderman, W. E.
5. Proceedings Fourth Berkeley Symp. Math. Statis. Probability, 1 Estimation with quadratic loss James, W.;Stein C.
6. Annals of Statistics v.3 Minimax estimation of location vectors for a wide class of densities Beger, J.
7. Annals of Statistics v.10 Differential geometry of curved exponential families, curvature and information loss Amari, S.
8. Journal of Multivariate Analysis v.32 On the best equivariant estimator of mean of a multivariate normal population Perron, F.;Giri, N.
9. Annals of Statistics v.17 Equivariant estimation in a model with ancillary statistics Kariya, T.
10. Handbook of Mathematical Functions Abramowitz, M.;Stegun, I.
11. Journal of the Royal Statistical Society, B v.24 Discussion of paper by C. Stein Lindley, D. V.
12. The Canadian Journal of Statistics v.10 An explicit formula for the risk of James-Stein estimators Egerton, M. F.;Laycock, P. J.