We study a class of tests of the randomness of Bernoulli sequences and their application to analyses of the human tendency to perceive streaks  as overly representative of positive dependence-the hot hand fallacy. In particular, we study tests 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 derive the asymptotic distributions of these test statistics and their permutation distributions under randomness and under general models of streakiness, which allows us to evaluate their local asymptotic power. The results are applied to revisit tests of the hot hand fallacy implemented on data from a basketball shooting experiment, whose conclusions are disputed by Gilovich, Vallone, and Tversky (1985) and Miller and Sanjurjo (2018a). We establish that the tests are insufficiently powered to distinguish randomness from alternatives consistent with the variation in NBA shootingpercentages. While multiple testing procedures reveal that one shooter can be inferred to exhibit shooting significantly inconsistent with randomness , we find that participants in a survey of basketball fans over-estimate an average player's streakiness, corroborating the empirical support for the hot hand fallacy .
We study a class of tests of the randomness 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 tests 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 derive the asymptotic distributions of these test statistics and their permutation distributions under randomness , under a set of general stationary processes, and under a Markov model of "streakiness", which allow us to evaluate their local asymptotic power. The results are applied to evaluate tests of randomness implemented on data from a basketball shooting experiment, whose conclusions are disputed by Gilovich, Vallone, and Tversky (1985) and Miller and Sanjurjo (2018a). We establish that substantially larger data sets are required to derive an informative measurement of the deviation from randomness in basketball shooting. Although multiple testing procedures reveal that one shooter in the experiment exhibits a shooting pattern significantly inconsistent with randomness - supplying strong evidence that basketball shooting is not random for all shooters all of the time - we find that the evidence against randomness 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 in Bernoulli sequences .
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of consecutive successes
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and under general models of streakiness, which allows
, under a set of general stationary processes, and under a Markov model of "streakiness", which allow
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revisit tests of the hot hand fallacy
evaluate tests of randomness
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the tests are insufficiently powered to distinguish randomness from alternatives consistent with the variation in NBA shootingpercentages. While
substantially larger data sets are required to derive an informative measurement of the deviation from randomness in basketball shooting. Although
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can be inferred to exhibit shooting
in the experiment exhibits a shooting pattern
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,
- supplying strong evidence that basketball shooting is not random for all shooters all of the time -
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participants in a survey of basketball fans over-estimate an average player's streakiness, corroborating the empirical support for the hot hand fallacy
the evidence against randomness 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 in Bernoulli sequences