On the optimality of coin-betting for mean estimation

TL;DR

This paper demonstrates the optimality of coin-betting strategies for testing the mean of bounded real variables, providing a comprehensive characterization of all valid e-variables and e-processes.

Contribution

It introduces a notion of optimal classes for e-variables and e-processes and proves the coin-betting formulation's optimality in this context.

Findings

Coin-betting formulation is optimal among e-variable-based testing methods.

Provides explicit characterization of all valid e-variables and e-processes.

Fully describes the set of admissible e-variables and e-processes for mean testing.

Abstract

We consider the problem of testing the mean of a bounded real random variable. We introduce a notion of optimal classes for e-variables and e-processes, and establish the optimality of the coin-betting formulation among e-variable-based algorithmic frameworks for testing and estimating the (conditional) mean. As a consequence, we provide a direct and explicit characterisation of all valid e-variables and e-processes for this testing problem. In the language of classical statistical decision theory, we fully describe the set of all admissible e-variables and e-processes, and identify the corresponding minimal complete class.