Whither jackknifing in stein-rule estimation

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Abstract
In multi-parameter ( multivariate ) estimation, the Stein rule provides minimax and admissible estimators , compromising generally on their unbiasedness. On the other hand, the primary aim of jack-knifing is to reduce the bias of an estimator ( without necessarily compromising on its efficacy ), and, at the same time, jackknifing provides an estimator of the sampling variance of the estimator as well. In shrinkage estimation ( where minimization of a suitably defined risk function is the basic goal ), one may wonder how far the bias-reduction objective of jackknifing incorporates the dual objective of minimaxity ( or admissibility ) and estimating the risk of the estimator ? A critical appraisal of this basic role of jackknifing in shrinkage estimation is made here. Restricted, semi-restricted and the usual versions of jackknifed shrinkage estimates are considered and their performance characteristics are studied . It is shown that for Pitman-type ( local ) alternatives, usually, jackkntfing fails to provide a consistent estimator of the ( asymptotic ) risk of the shrinkage estimator, and a degenerate asymptotic situation arises for the usual fixed alternative case.