Competitive betting improves test power but limits expected gains
Competitive optimality in testing by betting via Bell-Cover randomization
Computer Science and Game TheoryInformation Theory
Summary
The paper studies a way to improve statistical tests by randomizing betting amounts, inspired by a finance method that beats opponents half the time. The authors show this approach can outperform other methods in certain competitive settings but at a cost to the average success of the test. They also explain when and how randomizing bets helps or doesn't, especially when stopping tests early. This work clarifies trade-offs between maximizing test power and maintaining strong overall performance.
What this means in practice
- •For statistical data analysts: Use randomization techniques to enhance competitive power in hypothesis testing against composite null models.
- •For financial risk modelers: Incorporate Bell-Cover inspired betting randomization to refine wealth comparison methods under uncertainty and stopping rules.
A theory result. No direct application yet.
Authors
Aaditya Ramdas
Abstract
Bell and Cover showed that an investor who multiplies the initial unit of capital by an independent uniform random variable on $(0,2)$, and then uses the log-optimal portfolio, wins a head-to-head wealth comparison with probability at least one half against every independently randomized competitor. We explain very simply how this result transfers to testing by betting: for any composite null $\mathcal P$ and simple alternative $Q$, denoting $E^*$ as the corresponding numeraire e-variable, we show that $UE^*$ exceeds any other e-variable $E$ with probability at least half. Interestingly, we show that this competitive optimality result is actually equivalent to the numeraire inequality $\mathbb E_Q[E/E^*]\leq1$, and in general randomization only helps the numeraire and fails to improve the competitive advantage of an arbitrary e-variable. Under optional stopping with or without knowledge of $U$, we emphasize a key distinction between e-process validity and competitive optimality. We also show that competitive optimality comes at the price of expected log wealth and power: thresholding $UE^*$ at $1/α$ has sharp size at most $α/2$, but the factor of two actually disappears under optional stopping. Even after correcting for this factor of two, the test is dominated in conditional rejection probability by randomizing the testing threshold (randomized Markov's inequality). Thus, Bell-Cover randomization is optimal for a specific competitive objective, at the cost of others.