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 $m = o(n)$.

翻译：我们研究了估计两个随机变量联合概率质量函数的问题。具体而言，估计基于观测到的包含两个变量的 \(m\) 个样本和缺失一个固定变量的 \(n\) 个样本。我们采用具有 \(l^p_p\) 损失函数的极小极大框架，并证明当 \(p \ge 2\) 且处于 \(m = o(n)\) 的范围内时，单变量极小极大估计量的组合可实现具有最优一阶常数的极小极大风险。