通过Bell-Cover随机化的投注检验中的竞争最优性
Competitive optimality in testing by betting via Bell-Cover randomization
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中文总结 AI 辅助
本文证明Bell-Cover随机化使投注检验的e变量在头对头比较中获胜概率至少一半，等价于计价单位不等式，但以期望对数财富和功效为代价。
中文摘要 AI 辅助
Bell和Cover证明，一个投资者将初始单位资本乘以$(0,2)$上的独立均匀随机变量，然后使用对数最优投资组合，在与任何独立随机化竞争者的头对头财富比较中，获胜概率至少为二分之一。我们非常简单地解释了这一结果如何转移到投注检验：对于任何复合零假设$\mathcal P$和简单备择假设$Q$，记$E^*$为相应的计价单位e变量，我们证明$UE^*$超过任何其他e变量$E$的概率至少为一半。有趣的是，我们证明这一竞争最优性结果实际上等价于计价单位不等式$\mathbb E_Q[E/E^*]\leq1$，并且一般来说，随机化只对计价单位有帮助，而不能改善任意e变量的竞争优势。在可选停止下，无论是否知道$U$，我们强调e过程有效性与竞争最优性之间的关键区别。我们还证明，竞争最优性以期望对数财富和功效为代价：将$UE^*$阈值设为$1/\alpha$具有至多$\alpha/2$的锐利尺寸，但在可选停止下，因子二实际上消失。即使在修正这一因子二之后，该检验在条件拒绝概率上被随机化检验阈值（随机化马尔可夫不等式）所支配。因此，Bell-Cover随机化在特定竞争目标下是最优的，但以其他目标为代价。
英文摘要
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.
