Superefficiency from the Vantage Point of Computability

TL;DR

This paper explores the concept of superefficiency in statistical estimation through the lens of computability, showing that only computable points can be superefficient and that such points form a countable set.

Contribution

It introduces a computability-based framework to analyze superefficiency, extending Le Cam's classical measure-zero result to a countability result for computable estimators.

Findings

Superefficient points are only at computable parameters.

Sets of superefficient points are countable.

Superefficiency is constrained by computability considerations.

Abstract

In 1952 Lucien Le Cam announced his celebrated result that, for regular univariate statistical models, sets of points of superefficiency have Lebesgue measure zero. After reviewing the turbulent history of early studies of superefficiency, I suggest using the notion of computability as a tool for exploring the phenomenon of superefficiency. It turns out that only computable parameter points can be points of superefficiency for computable estimators. This algorithmic version of Le Cam's result implies, in particular, that sets of points of superefficiency not only have Lebesgue measure zero but are even countable.