Consider the following estimation problem: there are @math entities, each with an unknown parameter @math , and we observe @math independent random variables, @math , with @math Binomial @math . How accurately can one recover the "histogram" (i.e. cumulative density function) of the @math 's? While the empirical estimates would recover the histogram to earth mover distance @math (equivalently, @math distance between the CDFs), we show that, provided @math is sufficiently large, we can achieve error @math which is information theoretically optimal. We also extend our results to the multi-dimensional parameter case, capturing settings where each member of the population has multiple associated parameters. Beyond the theoretical results, we demonstrate that the recovery algorithm performs well in practice on a variety of datasets, providing illuminating insights into several domains, including politics, sports analytics, and variation in the gender ratio of offspring.
Abstract : If one observes the real random variables Xi, Xn independently normally distributed with unknown means xi...x in and variance 1, it is customary to estimate xi by Xi. If the loss is the sum of squares of the errors, this estimator is admissible for n or equal to 2, but inadmissible for n more than or equal to 3. Since the usual estimator is best among those which transform correctly under translation, any admissible estimator for n equals more than or equal to 3 involves an arbitrary choice. While the results of this paper are not in a form suitable for immediate practical application, the possible improvement over the usual estimator seems to be large enough to be of practical importance if n is large.