Aris Spanos

By Aris Spanos

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 θ in Θ” [∀θ∈Θ], irrespective of what the true value θ∗ [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 θ∗. The inappropriateness of the quantifier ∀θ∈Θ 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 θ∗, needs to be reassessed. The paper also contrasts loss‐based errors with traditional frequentist errors, arguing that the former are attached to θ, but the latter to the inference procedure itself.

Part of the book: Advances in Statistical Methodologies and Their Application to Real Problems