Statistical Decisions and Partial Identification: With Application to Boundary Discontinuity Design

TL;DR

This paper investigates how statistical decision theory can guide treatment assignment when the treatment effect is only partially identified, using a stylized model where $\hat{\mu} \sim N(\mu,\sigma)$ and $\mu^* \in [\mu-k,\mu+k]$. It derives minimax regret–based decision rules (MMR) that either ignore or partially incorporate identification uncertainty, yielding $1_{\{\hat{\mu}>0\}}$ as optimal for small identification gaps and a calibrated linear-rule with potential randomization for larger gaps, with the critical threshold $k^*<k$ solving $k^*/(2k) - 1/2 + \Phi(-k^*/\sigma) = 0$. The framework is then applied to a Boundary Discontinuity Design in educational subsidies (CTY25a), where a Lipschitz constraint on the heterogeneous treatment effect induces explicit bounds $\mu^*$ and $k$, and an MMR rule is shown to be nonrandomizing under a clear inequality involving the bound width and the variance $V$. The paper discusses practical leaps of faith in carrying the theory to application, reports a CTY25a–inspired numerical illustration, and lays out open challenges in scaling minimax rules and integrating partial identification with empirical decision problems.

Abstract

We are delighted to respond to the excellent surveys by Cattaneo et al. (2026) and Hirano (2026). Our discussion will attempt two things: first, we show how statistical decision theory can be applied to situations with partial identification; second, we connect the surveys' themes by applying these insights to an imagined policy experiment in one of Cattaneo et al.'s (2025) applications. To do so, we lay out a stylized scenario of statistical decision making under partial identification and, drawing on our own and others' earlier work, provide a complete solution for that scenario. We then apply these results to a hypothetical reduction (modelled on actual policies) in eligibility for educational subsidies. We will see that something of interest can be said, but also that bringing the theory to the application involves some leaps of faith and leaves some questions open. This leads to the final section, where we discuss what we see as the main open challenges in statistical decision theory under partial identification.

Figures (2)

• Figure 1: MMR treatment choice as function of $\hat{\mu}$; see Theorem \ref{['thm:1']}. Left: For small $k$ (or large $\sigma)$, model uncertainty is ignored. Right: For large $k$, there is randomization, but the recommended decision still effectively ignores model uncertainty for some signals $\hat{\mu}$ (with absolute value between $k^*$ and $k$) s.t. estimated bounds on the treatment effect do not identify its sign.
• Figure 2: Left: A policy experiment in the setting of CTY25a. Eligibility for treatment follows a double threshold rule; we consider increasing the first threshold. Right: Approximated (black, dashed) and exact (red, solid; compare the right panel of Figure \ref{['fig:T1']}) MMR rule from Theorem \ref{['thm:1']}. See fernandez2024epsilon for parameter values.

Theorems & Definitions (2)

• Theorem 1
• Proposition 1