We study a class of permutation tests of the randomness of a collection of Bernoulli sequences and their application to analyses of the human tendency to perceive streaks of consecutive successes as overly representative of positive dependence—the hot hand fallacy. In particular, we study permutation tests of the null hypothesis of randomness (i.e. that trials are i.i.d.) based on test statistics that compare the proportion of successes that directly follow $k$ consecutive successes with either the overall proportion of successes or the proportion of successes that directly follow $k$ consecutive failures. We characterize the asymptotic distributions of these test statistics and their permutation distributions under randomness, under a set of general stationary processes, and under a class of Markov chain alternatives, which allow us to derive their local asymptotic power. The results are applied to evaluate the empirical support for the hot hand fallacy provided by four controlled basketball shooting experiments. We establish that substantially larger data sets are required to derive an informative measurement of the deviation from randomness in basketball shooting. In one experiment, for which we were able to obtain data, multiple testing procedures reveal that one shooter exhibits a shooting pattern significantly inconsistent with randomness—supplying strong evidence that basketball shooting is not random for all shooters all of the time. However, we find that the evidence against randomness in this experiment is limited to this shooter. Our results provide a mathematical and statistical foundation for the design and validation of experiments that directly compare deviations from randomness with human beliefs about deviations from randomness and thereby constitute a direct test of the hot hand fallacy.

中文翻译：

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热手谬误中的不确定性：检测随机伯努利序列的条纹替代方案

我们研究了一类对伯努利序列集合的随机性的置换检验，并将它们应用于分析人类将连续成功的条纹视为过度代表积极依赖的倾向——热手谬误。特别是，我们研究了随机性零假设（即试验是独立同分布的）的置换检验，该检验基于检验统计量，该检验统计量将直接遵循 $k$ 连续成功的成功比例与成功的总体比例或成功的比例进行比较直接跟随 $k$ 连续失败。我们描述了这些测试统计量的渐近分布及其在随机性、一组一般平稳过程和一类马尔可夫链备选方案下的排列分布，这使我们能够推导出它们的局部渐近幂。将结果用于评估四个受控篮球投篮实验提供的热手谬误的经验支持。我们确定需要更大的数据集来获得对篮球投篮随机性偏差的信息测量。在一项我们能够获得数据的实验中，多个测试程序表明，一名射手的投篮模式与随机性显着不一致——这提供了强有力的证据，表明篮球投篮并非始终对所有射手都是随机的。然而，我们发现在这个实验中反对随机性的证据仅限于这个射手。