Rao-Blackwellized e-variables
Dante de Roos, Ben Chugg, Peter Grünwald, Aaditya Ramdas
TL;DR
The paper establishes a general Rao-Blackwellization principle for e-statistics, showing that conditioning an e-variable (or its sequential analogues) on a sufficient statistic can only improve, under any concave utility, the expected value across the parameter space. It extends the result to log-utility with weaker existence conditions, to e-processes via sufficient filtrations, and to compound and asymptotic e-variables, providing a versatile theoretical tool. Concrete applications demonstrate how the approach recovers optimal e-variables in fixed-design linear regression and Pareto data, and how permutation-based averaging can enhance GRO/GROW properties for i.i.d. data. Collectively, the results offer a unifying framework to design and analyze powerful e-statistics across static and sequential testing contexts.
Abstract
We show that for any concave utility, the expected utility of an e-variable can only increase after conditioning on a sufficient statistic. The simplest form of the result has an extremely straightforward proof, which follows from a single application of Jensen's inequality. Similar statements hold for compound e-variables, asymptotic e-variables, and e-processes. These results echo the Rao-Blackwell theorem, which states that the expected squared error of an estimator can only decrease after conditioning on a sufficient statistic. We provide several applications of this insight, including a simplified derivation of the log-optimal e-variable for linear regression with known variance.