Empirical Likelihood for Nonsmooth Functionals

1. Problem Statement

The paper addresses a critical gap in statistical inference for policy evaluation, specifically the estimation of the optimal value (the maximum expected reward) among a set of candidate policies.

• The Functional: The target of interest is $\tau(P_0) = \max_{1 \le j \le J} \theta_j(P_0)$, where $\theta_j$ represents the value of policy $\pi_j$.
• The Challenge (Nonsmoothness): The "max" functional is nonsmooth (specifically, Hadamard directionally differentiable but not fully differentiable) when multiple policies are tied for the optimal value or are near-tied.
• Limitations of Existing Methods:
  • Standard Empirical Likelihood (EL): Classical EL theory (e.g., Owen, Hjort et al.) relies on smoothness assumptions (Taylor expansions of dual Lagrange multipliers). When applied to nonsmooth functionals, standard EL fails to provide valid confidence intervals.
  • Projected Joint EL: Constructing a joint confidence region for all policy values and projecting onto the maximum is valid but overly conservative. It incurs a dimensionality penalty (using a $\chi^2_J$ critical value for a scalar target), resulting in bounds that are unnecessarily wide (scaling with $\sqrt{J}$).
  • Selected-Policy EL: Estimating the single "best" policy in the sample and constructing a confidence interval for it ignores selection uncertainty. This leads to severe undercoverage (miscalibration) when policies are near-tied, precisely the regime where rigorous inference is most needed.
  • Standard Bootstrap: The ordinary bootstrap fails for nonsmooth functionals because it linearizes the problem, estimating the level set as a hyperplane rather than the correct tangent cone, leading to incorrect asymptotic distributions.

2. Methodology

The author proposes a Score-Profile Empirical Likelihood framework that operates directly on the score vectors derived from semiparametric estimation (specifically Augmented Inverse Probability Weighting or AIPW scores), avoiding the need to refit nuisance models during bootstrapping.

A. Geometric Reduction (Primal Approach)

Instead of the traditional dual approach (expanding Lagrange multipliers), the paper introduces a primal geometric proof strategy:

• The Reduction: The profile empirical likelihood statistic is shown to be equivalent to the squared Mahalanobis distance between the sample mean of the score vectors ($\bar{X}_n$) and the local level set of the target functional.
• The Level Set Geometry:
  • Smooth Case (Unique Optimum): If one policy is strictly optimal, the level set converges to a hyperplane. The distance follows a standard $\chi^2_1$ distribution.
  • Nonsmooth Case (Ties): If multiple policies are tied, the level set converges to a tangent cone. The distance follows a non-pivotal distribution (a weighted mixture of $\chi^2$ distributions, denoted $\bar{\chi}^2$), dependent on the cone's geometry and the covariance structure.

B. Bootstrap Calibration

Since the asymptotic distribution in the nonsmooth case depends on unknown geometry (the set of tied policies), the paper develops a Corrected Multiplier Bootstrap:

1. Active Set Estimation: The method estimates the set of near-optimal policies ($\hat{J}_n$) using a thresholding rule based on the sample gaps and covariance.
2. Cone Estimation: An estimated tangent cone ($\hat{C}_n$) is constructed based on $\hat{J}_n$.
3. Multiplier Bootstrap: The procedure generates bootstrap samples of the score vectors using multipliers, computes the Mahalanobis distance to the estimated cone $\hat{C}_n$, and uses the empirical quantile of these distances as the critical value.
4. Computational Efficiency: Because the method operates on pre-computed AIPW scores, the bootstrap loop requires only $O(nJ)$ operations per draw (matrix multiplication) and does not require refitting the nuisance models (propensity scores or outcome regressions) inside the bootstrap loop.

C. Closed-Form Bounds

For the max functional, the lower confidence bound can be expressed as a simplex optimization problem:
$$L_n(c) = \max_{w \in \Delta_J} \left\{ w^\top \bar{X}_n - \sqrt{\frac{c}{n}} \sqrt{w^\top \hat{\Sigma}_n w} \right\}$$
This formulation allows for efficient computation and reveals a "diversification" effect: the optimizer can mix weights across near-tied policies to reduce variance, yielding tighter bounds than pointwise maximum methods.

3. Key Contributions

1. Primal Geometric Framework: Replaces the dual Lagrange-multiplier expansion with a geometric reduction of the profile statistic to a distance-to-set problem. This avoids the technical obstructions of nonsmoothness in dual expansions.
2. Adaptive Inference: The method automatically adapts to the geometry of the problem (hyperplane vs. cone) without requiring the user to verify the uniqueness of the optimum a priori.
3. Corrected Multiplier Bootstrap: Derives a consistent bootstrap procedure for nonsmooth functionals that estimates the tangent cone from data, correcting the failure of the ordinary bootstrap.
4. Computational Scalability: By decoupling the bootstrap from nuisance model refitting, the method offers orders-of-magnitude speedups compared to standard semiparametric EL bootstraps (e.g., Hjort et al.), making it feasible for large-scale policy evaluation.
5. Portfolio Diversification in Inference: The simplex formulation exploits low correlation between near-optimal policies to produce tighter confidence bounds than pointwise methods (like Fang and Santos [7]).

4. Results

The paper validates the method through extensive simulations and theoretical proofs:

• Coverage Accuracy:
  • Unique Optimum: The method achieves nominal coverage (95%) with tight bounds, avoiding the $\sqrt{J}$ penalty of projected joint EL.
  • Near-Ties: The method maintains valid coverage where standard "selected-policy" Wald intervals fail (dropping to <90% coverage) and projected joint EL is overly conservative.
• Tightness: The proposed bounds are significantly tighter (roughly 3x in some semiparametric settings) than projected joint EL bounds while maintaining valid coverage.
• Diversification Effect: In scenarios with multiple near-optimal policies that are weakly correlated, the simplex-based bound is 5–10% tighter than the Fang and Santos [7] pointwise-max bound.
• Computation: The score-level bootstrap is approximately 460x faster than a full refit bootstrap in semiparametric settings.

5. Significance

This work extends the applicability of Empirical Likelihood to a class of problems central to modern data-driven decision-making (e.g., personalized medicine, dynamic pricing) where the "best" option is often ambiguous or tied.

• Theoretical: It resolves the long-standing issue of applying EL to Hadamard directionally differentiable functionals, providing a unified framework for both smooth and nonsmooth regimes.
• Practical: It offers a computationally feasible, statistically rigorous tool for policymakers to quantify the value of the best available strategy without falling into the traps of selection bias or excessive conservatism.
• Methodological: The shift from dual expansions to primal geometric projections offers a new paradigm for analyzing complex, nonsmooth statistical functionals.