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 <tex>$m$</tex> samples containing both variables and <tex>$n$</tex> samples missing one fixed variable. We adopt the minimax framework with <tex>$l_{p}^{p}$</tex> loss functions. Recent work established that univariate minimax estimator combinations achieve minimax risk with the optimal first-order constant for <tex>$p\geq 2$</tex> in the regime <tex>$m=o(n)$</tex>, questions remained for <tex>$p\leq 2$</tex> and various <tex>$f$</tex> -divergences. In our study, we affirm that these composite estimators are indeed minimax optimal for <tex>$l_{p}^{p}$</tex> loss functions, specifically for the range <tex>$1\leq p\leq 2$</tex>, including the critical <tex>$l_{1}$</tex> loss. Additionally, we ascertain their optimality for a suite of <tex>$f$</tex> -divergences, such as KL, <tex>$\chi^{2}$</tex>, Squared Hellinger, and Le Cam divergences.