  Consider a setting with $N$ independent individuals, each with an unknown
parameter, $p_i \in [0, 1]$ drawn from some unknown distribution $P^\star$.
After observing the outcomes of $t$ independent Bernoulli trials, i.e., $X_i
\sim \text{Binomial}(t, p_i)$ per individual, our objective is to accurately
estimate $P^\star$. 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 $t
\ll N$, 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
$\mathcal{O}(\frac{1}{t})$ for $t < c\log{N}$, with regards to the earth
mover's distance (between the estimated and true distributions). More
generally, in an exponentially large interval of $t$ beyond $c \log{N}$, the
MLE achieves the minimax error bound of $\mathcal{O}(\frac{1}{\sqrt{t\log
N}})$. In contrast, regardless of how large $N$ is, the naive "plug-in"
estimator for this problem only achieves the sub-optimal error of
$\Theta(\frac{1}{\sqrt{t}})$.
