Detecting the Hot Hand: Tests of Randomness Against Streaky Alternatives in Bernoulli Sequences
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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.
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