Abstract:

Consider a setting with N independent individuals, each with an unknown parameter, pi∈[0,1] drawn from some unknown distribution P⋆. After observing the outcomes of t independent Bernoulli trials, i.e., Xi∼Binomial(t,pi) per individual, our objective is to accurately estimate P⋆. This problem arises in numerous domains, including the social sciences, psychology, health-care, and biology, where the size of the population under study is usually large while the number of observations per individual is often limited. Our main result shows that, in the regime where t≪N, the maximum likelihood estimator (MLE) is both statistically minimax optimal and efficiently computable. Precisely, for sufficiently large N, the MLE achieves the information theoretic optimal error bound of (1t) for t

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