Risk and Admissibility

The risk function records a rule's expected loss at every parameter value; admissibility asks whether any other rule does at least as well everywhere and better somewhere.

Definition

The risk function of a decision rule is the expected loss as a function of the parameter; a rule is inadmissible if some other rule has risk no larger for all parameter values and strictly smaller for at least one, and admissible if no such rule exists.

Scope

This topic covers loss functions and the risk function, the partial ordering of rules by risk dominance, the definitions of admissible and inadmissible rules, the inadmissibility of the sample mean in three or more dimensions as the central example, methods for proving admissibility through Bayes and limiting-Bayes arguments and Stein's identity, and the relationship between admissibility and unbiasedness.

Core questions

• How does the risk function summarize a rule's performance across the parameter space?
• What does it mean for one rule to dominate another, and hence for a rule to be inadmissible?
• Why is the sample mean inadmissible in three or more dimensions under squared-error loss?
• How are Bayes and limiting-Bayes arguments used to prove admissibility?

Key theories

Risk dominance and admissibility

A rule is inadmissible when another rule has uniformly no greater and somewhere strictly smaller risk; admissible rules are those that cannot be uniformly improved, the minimal optimality requirement.

Stein's inadmissibility

Under squared-error loss the usual estimator of a multivariate normal mean is inadmissible in three or more dimensions, dominated by shrinkage estimators, a result proved using Stein's identity.

Clinical relevance

Recognizing that a familiar estimator can be inadmissible justifies the routine use of shrinkage and regularization in high-dimensional prediction, where pulling estimates toward a common center provably lowers total risk compared with treating each coordinate separately.

History

Wald introduced risk and admissibility in the 1940s. Stein's 1956 proof that the multivariate normal mean estimator is inadmissible in three or more dimensions overturned intuition and, with the James-Stein estimator of 1961, made admissibility a central concern.

Key figures

• Abraham Wald
• Charles Stein
• David Blackwell
• James O. Berger

Seminal works

• lehmannCasella1998

Frequently asked questions

If a rule is admissible, is it the best rule?

No. Admissibility only rules out being uniformly beaten; many admissible rules are mediocre and a good rule can be inadmissible, so admissibility is a necessary but far from sufficient condition for optimality.

Why does dimension three matter for Stein's result?

The inadmissibility of the sample mean under squared-error loss holds in three or more dimensions but not in one or two; below three, shrinkage cannot uniformly improve on the sample mean.