Some data-analytic modifications to Bayes-Stein estimation

Summary The usual Bayes-Stein shrinkages of maximum likelihood estimates towards a common value may be refined by taking fuller account of the locations of the individual observations. Under a Bayesian formulation, the types of shrinkages depend critically upon the nature of the common distribution assumed for the parameters at the second stage of the prior model. In the present paper this distribution is estimated empirically from the data, permitting the data to determine the nature of the shrinkages. For example, when the observations are located in two or more clearly distinct groups, the maximum likelihood estimates are roughly speaking constrained towards common values within each group. The method also detects outliers; an extreme observation will either the regarded as an outlier and not substantially adjusted towards the other observations, or it will be rejected as an outlier, in which case a more radical adjustment takes place. The method is appropriate for a wide range of sampling distributions and may also be viewed as an alternative to standard multiple comparisons, cluster analysis, and nonparametric kernel methods.