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Improved shrinkage estimators for the mean vector of a scale mixture of normals with unknown variance
Author(s) -
Bravo Gina,
Macgibbon Brenda
Publication year - 1988
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.2307/3314730
Subject(s) - mathematics , estimator , shrinkage , shrinkage estimator , minimax , random variate , diagonal , covariance matrix , matrix (chemical analysis) , covariance , statistics , normal distribution , multivariate normal distribution , quadratic equation , variance (accounting) , scale (ratio) , mathematical optimization , bias of an estimator , random variable , minimum variance unbiased estimator , multivariate statistics , geometry , materials science , accounting , physics , quantum mechanics , business , composite material
The problem of estimating the mean θ of a not necessarily normal p ‐variate ( p > 3) distribution with unknown covariance matrix of the form σ 2 A (A a known diagonal matrix) on the basis of n i > 2 observations on each coordinate X t (1 < i < p ) is considered. It is argued that the class of scale (or variance) mixtures of normal distributions is a reasonable class to study. Assuming the loss function is quadratic, a large class of improved shrinkage estimators is developed in the case of a balanced design. We generalize results of Berger and Strawderman for one observation in the known‐variance case. This methodology also permits the development of a new class of minimax shrinkage estimators of the mean of a p‐variate normal distribution for an unbalanced design. Numerical calculations show that the improvements in risk can be substantial.