  In this paper, we develop a functional differentiability approach for solving
statistical optimal allocation problems. We first derive Hadamard
differentiability of the value function through a detailed analysis of the
general properties of the sorting operator. Central to our framework are the
concept of Hausdorff measure and the area and coarea integration formulas from
geometric measure theory. Building on our Hadamard differentiability results,
we demonstrate how the functional delta method can be used to directly derive
the asymptotic properties of the value function process for binary constrained
optimal allocation problems, as well as the two-step ROC curve estimator.
Moreover, leveraging profound insights from geometric functional analysis on
convex and local Lipschitz functionals, we obtain additional generic Fr\'echet
differentiability results for the value functions of optimal allocation
problems. These compelling findings motivate us to study carefully the first
order approximation of the optimal social welfare. In this paper, we then
present a double / debiased estimator for the value functions. Importantly, the
conditions outlined in the Hadamard differentiability section validate the
margin assumption from the statistical classification literature employing
plug-in methods that justifies a faster convergence rate.
