Families of Minimax Estimators of the Mean of a Multivariate Normal Distribution
Efron, Bradley ; Morris, Carl
Ann. Statist., Tome 4 (1976) no. 1, p. 11-21 / Harvested from Project Euclid
Ever since Stein's result, that the sample mean vector $\mathbf{X}$ of a $k \geqq 3$ dimensional normal distribution is an inadmissible estimator of its expectation $\mathbf{\theta}$, statisticians have searched for uniformly better (minimax) estimators. Unbiased estimators are derived here of the risk of arbitrary orthogonally-invariant and scale-invariant estimators of $\mathbf{\theta}$ when the dispersion matrix $\sum$ of $\mathbf{X}$ is unknown and must be estimated. Stein obtained this result earlier for known $\mathbf{\sum}$. Minimax conditions which are weaker than any yet published are derived by finding all estimators whose unbiased estimate of risk is bounded uniformly by $k$, the risk of $\mathbf{X}$. One sequence of risk functions and risk estimates applies simultaneously to the various assumptions about $\mathbf{\sum}$, resulting in a unified theory for these situations.
Publié le : 1976-01-14
Classification:  Estimation,  minimax estimators,  risk of invariant estimators,  mean of a multivariate normal distribution,  Stein's estimator,  62F10,  62C99
@article{1176343344,
     author = {Efron, Bradley and Morris, Carl},
     title = {Families of Minimax Estimators of the Mean of a Multivariate Normal Distribution},
     journal = {Ann. Statist.},
     volume = {4},
     number = {1},
     year = {1976},
     pages = { 11-21},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1176343344}
}
Efron, Bradley; Morris, Carl. Families of Minimax Estimators of the Mean of a Multivariate Normal Distribution. Ann. Statist., Tome 4 (1976) no. 1, pp.  11-21. http://gdmltest.u-ga.fr/item/1176343344/