A Warm Bath for the Hot Hand Methodology

Existing tests for the ``hot hand'' in sports can only ask whether streaks carry no information, because they typically rely on permutation tests. The null hypotheses associated with these permutation tests admit no parameter for the size of a hot hand effect. We instead model the hot hand as a log-odds shift $\delta$ in the hit probability after a streak, alongside a nuisance baseline skill $\eta$. Conditioning on two counts, the hits and the trials that follow a streak, removes $\eta$ while leaving $\delta$ identified, and extends testing validity to strictly more laws than exchangeability. We measure evidence against hypotheses on $\delta$, including the one-sided statements $\delta \le \delta_0$ and $\delta \ge \delta_0$ that prior work cannot express, using e-values, whose evidence multiplies across independent shooters without a multiple-testing correction. We verify validity numerically and apply the method to basketball data, where merging shooters turns weak sequences into a usable verdict.