In this paper, we develop a functional differentiability approach for solving statistical optimal allocation problems. We derive Hadamard differentiability of the value functions through analyzing the properties of the sorting operator using tools from geometric measure theory. Building on our Hadamard differentiability results, we apply the functional delta method to obtain the asymptotic properties of the value function process for the binary constrained optimal allocation problem and the plug-in ROC curve estimator. Moreover, the convexity of the optimal allocation value functions facilitates demonstrating the degeneracy of first order derivatives with respect to the policy. We then present a double / debiased estimator for the value functions. Importantly, the conditions that validate Hadamard differentiability justify the margin assumption from the statistical classification literature for the fast convergence rate of plug-in methods.

翻译：本文提出了一种函数可微性方法来解决统计最优分配问题。通过运用几何测度论工具分析排序算子的性质，我们推导出值函数的Hadamard可微性。基于Hadamard可微性结果，我们应用函数delta方法获得了二元约束最优分配问题的值函数过程以及插件ROC曲线估计量的渐近性质。此外，最优分配值函数的凸性有助于证明其关于策略的一阶导数退化性。随后我们提出了值函数的双重/去偏估计量。值得注意的是，验证Hadamard可微性的条件为统计分类文献中关于插件方法快速收敛速率的边界假设提供了理论依据。