We consider the problem of testing for randomness against streaky alternatives in Bernoulli sequences . In particular, we study tests of randomness (i.e., that trials are i.i.d.) which choose as test statistics (i) the difference between the proportions of successes that directly follow k consecutive successes and k consecutive failures or (ii) the difference between the proportion of successes following k consecutive successes and the proportion of successes . The asymptotic distributions of these test statistics and their permutation distributions are derived 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).  While multiple testing procedures reveal that one shooter can be inferred to exhibit shooting significantly inconsistent with randomness, supporting the existence of positive dependence in basketball shooting, we find that participants in a survey of basketball players 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 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 shooting percentages. 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.
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consider the problem of testing for randomness against streaky alternatives in Bernoulli sequences
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
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which choose as test statistics (i) the difference between the proportions
based on test statistics that compare the proportion
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and k consecutive failures or (ii) the difference between the
with either the overall
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following k consecutive successes and
or
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. The
that directly follow k consecutive failures. We derive the
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are derived
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We establish that the tests are insufficiently powered to distinguish randomness from alternatives consistent with the variation in NBA shooting percentages.
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supporting the existence of positive dependence in basketball shooting,
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players
fans