On Semi-Supervised Estimation of Distributions

Abstract

We study the problem of estimating the joint probability mass function (pmf) over two random variables. In particular, the estimation is based on the observation of $m$ samples containing both variables and $n$ samples missing one fixed variable. We adopt the minimax framework with $l^p_p$ loss functions, and we show that the composition of uni-variate minimax estimators achieves minimax risk with the optimal first-order constant for $p \ge 2$, in the regime $m = o(n)$.