We investigate the asymptotic relation between likelihood ratios and p-values. We do that in a setting in which exact computations are possible: a coin-tossing context where the hypotheses of interest address the success probability of the coin. We obtain exact asymptotic results and conclude that the p-value scales very differently than the likelihood ratio. We also investigate the p-value of the likelihood ratio, that is, the probability of finding a more extreme likelihood ratio under the various hypotheses. Here we also find explicit asymptotic relations, with similar conclusions. Finally, we study the expected value of the likelihood ratio in an optional stopping context. Our results imply, for instance, that in a coin-tossing context, a p-value of 0.05 cannot correspond to an actual likelihood ratio larger than 6.8.

翻译：我们研究了似然比与p值之间的渐近关系。研究在一个可进行精确计算的情境下展开：以抛硬币为背景，关注假设检验中硬币成功概率的问题。我们获得了精确的渐近结果，并得出结论：p值的尺度变化与似然比存在显著差异。同时，我们考察了似然比的p值，即在各种假设下找到更极端似然比的概率。在此我们也发现了显式的渐近关系，并得出类似结论。最后，我们研究了可选停止背景下似然比的期望值。我们的研究结果表明，例如在抛硬币情境中，0.05的p值对应的实际似然比不可能超过6.8。