  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
$m = o(n)$.
