Truthful and Admissible Principal--Agent Testing with Nuisance Parameters

Abstract

Statistical contracts reward experimenters according to the evidence produced by a costly study, so the reward rule influences both who participates and what they report. Principal--agent testing theory explains how to prevent ineffective proposals from being profitable. We ask when such contracts can also elicit honest parameter reports if the decision concerns a mean effect but the variance or distributional shape is unknown. Under risk neutrality and a fixed participation cost, we construct menus of e-values --- nonnegative normalized rewards with expected value at most one under every null --- such that (i) agents who know they are null have no incentive to participate, (ii) every alternative type earns positive expected profit and is thus incentivized to participate, (iii) truthful reporting uniquely maximizes expected their reward, (iv) and each contract is admissible: its payouts cannot be pointwise improved while retaining null validity. For Gaussian observations, we treat separately agents who know both mean and variance and agents who know only their mean; in the latter case, truthfulness holds at every variance without requiring a nuisance prior. Multi-sample extensions preserve these incentives and can reward accumulating evidence. Beyond Gaussian models, we show that two independent nonnegative observations suffice to elicit any finite mean above a positive null threshold, without any further distributional knowledge, but one observation cannot combine strict mean elicitation with individual admissibility on this class. Finally, we distinguish the existence of useful tests from the identification needed for truthful contracts. Boundedness is not required, and finitely many null constraints do not by themselves guarantee admissible elicitation.