Statistical Inference of Optimal Allocations I: Regularities and their Implications

TL;DR

This paper introduces a functional differentiability framework for analyzing optimal allocation problems, deriving asymptotic properties and proposing a debiased estimator, with implications for statistical classification and resource allocation.

Contribution

It develops a novel functional differentiability approach for optimal allocation, deriving Hadamard differentiability and applying the delta method to obtain asymptotic results.

Findings

Hadamard differentiability of value functions established

Asymptotic properties of estimators derived

Debiased estimator for value functions proposed

Abstract

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.