🤖 AI Summary
Existing sequential mean testing methods struggle to characterize the higher-order asymptotic behavior of stopping times, particularly lacking second-order information-theoretic optimality analyses under bounded distributions. This work proposes a sequential test based on the KL_inf statistic, which not only achieves first-order asymptotic optimality but also establishes, for the first time, a central limit theorem for this statistic. Consequently, the stopping time—after appropriate centering and scaling by √log(1/α)—converges in distribution to a Gaussian limit. This result provides a refined second-order characterization of sequential test performance under bounded distributions, with theoretical proof showing convergence to a normal distribution having an explicit variance. Numerical experiments corroborate the accuracy of this second-order asymptotic analysis.
📝 Abstract
We revisit the problem of sequentially testing the mean of bounded distributions in a level-$α$ power-one framework. We study a $\mathrm{KL_{inf}}$-based sequential test that is known to attain the information-theoretic lower bound on the expected stopping time with exact constants as $α\to 0$. Going beyond first-order asymptotics, we establish a central limit theorem (CLT) for the stopping time of this test. Our analysis proceeds in two steps. First, we prove a novel CLT for the $\mathrm{KL_{inf}}$ statistic itself, characterizing its fluctuations around its deterministic limit. We then leverage this result to show that the stopping time, centered appropriately and scaled by $\sqrt{\log(1/α)}$, converges in distribution to a Gaussian limit with an explicit variance. This yields a second-order characterization of an asymptotically optimal sequential test for bounded distributions. Finally, we present numerical experiments that corroborate our theoretical findings.