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Empirical Likelihood for Nonsmooth Functionals

Hongseok Namkoong

Abstract

Empirical likelihood is an attractive inferential framework that respects natural parameter boundaries, but existing approaches typically require smoothness of the functional and miscalibrate substantially when these assumptions are violated. For the optimal-value functional central to policy evaluation, smoothness holds only when the optimum is unique -- a condition that fails exactly when rigorous inference is most needed where more complex policies have modest gains. In this work, we develop a bootstrap empirical likelihood method for partially nonsmooth functionals. Our analytic workhorse is a geometric reduction of the profile likelihood to the distance between the score mean and a level set whose shape (a tangent cone given by nonsmoothness patterns) determines the asymptotic distribution. Unlike the classical proof technology based on Taylor expansions on the dual optima, our geometric approach leverages properties of a deterministic convex program and can directly apply to nonsmooth functionals. Since the ordinary bootstrap is not valid in the presence of nonsmoothness, we derive a corrected multiplier bootstrap approach that adapts to the unknown level-set geometry.

Empirical Likelihood for Nonsmooth Functionals

Abstract

Empirical likelihood is an attractive inferential framework that respects natural parameter boundaries, but existing approaches typically require smoothness of the functional and miscalibrate substantially when these assumptions are violated. For the optimal-value functional central to policy evaluation, smoothness holds only when the optimum is unique -- a condition that fails exactly when rigorous inference is most needed where more complex policies have modest gains. In this work, we develop a bootstrap empirical likelihood method for partially nonsmooth functionals. Our analytic workhorse is a geometric reduction of the profile likelihood to the distance between the score mean and a level set whose shape (a tangent cone given by nonsmoothness patterns) determines the asymptotic distribution. Unlike the classical proof technology based on Taylor expansions on the dual optima, our geometric approach leverages properties of a deterministic convex program and can directly apply to nonsmooth functionals. Since the ordinary bootstrap is not valid in the presence of nonsmoothness, we derive a corrected multiplier bootstrap approach that adapts to the unknown level-set geometry.

Paper Structure

This paper contains 42 sections, 13 theorems, 107 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivating application
  3. Why are ties relevant?
  4. Limitations of existing EL approaches
  5. Contributions
  6. Related work
  7. Classical empirical likelihood.
  8. Semiparametric empirical likelihood.
  9. Debiased inference and policy evaluation.
  10. Inference for nonsmooth functionals.
  11. Score-profile framework and geometric reduction
  12. Score regularity
  13. Geometric reduction
  14. Level sets to limiting distributions
  15. Smooth Statistical Functionals
  16. ...and 27 more sections

Key Result

Theorem 1

Under Assumption ass:score, where $d_A^2(z,S) := \inf_{v \in S} (z-v)^\top A^{-1}(z-v)$ for positive definite $A$ and closed $S \subset \mathbb{R}^d$.

Figures (3)

  • Figure 1: Coverage and average shortfall of one-sided 95% lower bounds as a function of $J$, with $n = 500$ and a unique optimum. The projected joint approach pays a dimension penalty that grows as $\sqrt J$. Our method avoids this entirely.
  • Figure 2: Coverage of one-sided 95% lower bounds as a function of tie multiplicity $k$ ($J = 20$, $n = 500$ and $n = 1{,}000$). The selected-policy method undercovers sharply at ties; the projected joint EL method overcovers regardless; our method maintains near-nominal coverage by adapting to the cone geometry.
  • Figure 3: Our method vs. FangSa19 as the pairwise correlation $\rho$ among $k = 3$ tied policies varies ($J = 10$, $n = 2{,}000$, $5{,}000$ repetitions). (a) Both methods maintain near-nominal coverage across the range; our method matches or exceeds FangSa19 coverage throughout. (b) Our shortfall drops sharply as $\rho$ decreases because the simplex optimizer diversifies across tied policies; the FangSa19 shortfall is relatively flat.

Theorems & Definitions (13)

  • Theorem 1: Geometric reduction
  • Proposition 2: Level-set structure of the max functional
  • Theorem 3: Distance-to-set limit
  • Corollary 1: Smooth limit
  • Proposition 4: Lower bound via simplex optimization
  • Proposition 5: Inflation factor
  • Theorem 6: Ordinary score bootstrap
  • Corollary 2: Distance-to-cone limit
  • Proposition 7: Face decomposition
  • Proposition 8: Validity of the active-set estimator
  • ...and 3 more