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Statistical Inference of Optimal Allocations I: Regularities and their Implications

Kai Feng, Han Hong, Denis Nekipelov

TL;DR

The paper develops a functional differentiability framework for statistical optimal allocation problems by proving Hadamard differentiability of the value function with respect to prediction-augmented allocations using the sorting operator. It leverages area and coarea formulas and Hausdorff measure to express derivatives as surface integrals over level sets, enabling a functional delta method to derive asymptotics for binary constrained allocations and plug-in ROC curves, with bootstrap validity. Convexity in the value function leads to degeneracy of first-order derivatives, motivating Neyman-orthogonal and debiased estimators that yield fast convergence rates, including second-order Hadamard differentiability for multiclass settings. The framework extends to multiclass allocations and provides envelope-like results for the social welfare potential, supported by simulations and real-data applications (e.g., voting policy gains) that demonstrate reliable inference and competitive performance against established approaches. Overall, the work connects geometric measure theory with modern causal inference and policy learning, delivering practical, robust tools for high-dimensional, nonsmooth optimal allocation problems.

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.

Statistical Inference of Optimal Allocations I: Regularities and their Implications

TL;DR

The paper develops a functional differentiability framework for statistical optimal allocation problems by proving Hadamard differentiability of the value function with respect to prediction-augmented allocations using the sorting operator. It leverages area and coarea formulas and Hausdorff measure to express derivatives as surface integrals over level sets, enabling a functional delta method to derive asymptotics for binary constrained allocations and plug-in ROC curves, with bootstrap validity. Convexity in the value function leads to degeneracy of first-order derivatives, motivating Neyman-orthogonal and debiased estimators that yield fast convergence rates, including second-order Hadamard differentiability for multiclass settings. The framework extends to multiclass allocations and provides envelope-like results for the social welfare potential, supported by simulations and real-data applications (e.g., voting policy gains) that demonstrate reliable inference and competitive performance against established approaches. Overall, the work connects geometric measure theory with modern causal inference and policy learning, delivering practical, robust tools for high-dimensional, nonsmooth optimal allocation problems.

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.
Paper Structure (20 sections, 19 theorems, 182 equations, 3 tables, 3 algorithms)

This paper contains 20 sections, 19 theorems, 182 equations, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Literature
  3. Hadamard differentiability
  4. Hausdorff measure and integration formulas
  5. Hadamard derivatives of the sorting operator
  6. Binary treatment allocation and the ROC curve
  7. Binary constrained allocation
  8. The ROC curve
  9. Fast convergence rates
  10. Second order Hadamard differentiability
  11. Multi discrete allocation problem
  12. Inference on social welfare potential function
  13. Simulation and empirical application
  14. Confidence intervals and hypothesis testing of ROC curves
  15. Real data application and simulation
  16. ...and 5 more sections

Key Result

Theorem 3.1.1

Area formula Let $E \subset \mathbb{R}^{k}$ be an open set, $k \in \{1, 2, \ldots, n\}$, $\psi: E \mapsto \mathbb{R}^{n}$ be a $C^1$ or Lipschitz function. Then for all measurable $S \subset E$, If a measurable function $f: S \mapsto \mathbb{R}$ is nonnegative or if the left hand side of area formula integral is finite, then the following equality holds,

Theorems & Definitions (29)

  • Definition 3.1.1
  • Definition 3.1.2
  • Definition 3.1.3
  • Theorem 3.1.1
  • Theorem 3.1.2
  • Remark 3.1.1
  • Definition 3.2.1
  • Remark 3.2.1
  • Definition 3.2.2
  • Theorem 3.2.1
  • ...and 19 more