Why the Decision-Theoretic Perspective Misrepresents Frequentist Inference: Revisiting Stein's Paradox and Admissibility

The primary objective of this paper is to make a case that R.A. Fisher's objections to the decision-theoretic framing of frequentist inference are not without merit. It is argued that this framing is congruent with the Bayesian but incongruent with the frequentist approach; it provides the former with a theory of optimal inference but misrepresents the optimality theory of the latter. Decision-theoretic and Bayesian rules are considered optimal when they minimize the expected loss "for all possible values of ? in ?" [????], irrespective of what the true value ?* [state of Nature] happens to be; the value that gave rise to the data. In contrast, the theory of optimal frequentist inference is framed entirely in terms of the capacity of the procedure to pinpoint ?*. The inappropriateness of the quantifier ???? calls into question the relevance of admissibility as a minimal property for frequentist estimators. As a result, the pertinence of Stein's paradox, as it relates to the capacity of frequentist estimators to pinpoint ?*, needs to be reassessed. The paper also contrasts loss-based errors with traditional frequentist errors, arguing that the former are attached to ?, but the latter to the inference procedure itself.