Empirical Likelihood for Random Sets

Abstract

In many statistical applications, the observed data take the form of sets rather than points. Examples include bracket data in survey analysis, tumor growth and rock grain images in morphology analysis, and noisy measurements on the support function of a convex set in medical imaging and robotic vision. Additionally, in studies of treatment effects, researchers often wish to conduct inference on nonparametric bounds for the effects which can be expressed by means of random sets. This article develops the concept of nonparametric likelihood for random sets and its mean, known as the Aumann expectation, and proposes general inference methods by adapting the theory of empirical likelihood. Several examples, such as regression with bracket income data, Boolean models for tumor growth, bound analysis on treatment effects, and image analysis via support functions, illustrate the usefulness of the proposed methods. Supplementary materials for this article are available online.

KEYWORDS:

• Empirical likelihood
• Random set
• Treatment effect

Supplementary Materials

The online supplement contains additional appendices for the article.

Acknowledgments

The authors would like to thank Hiroaki Kaido and the Associate Editor for helpful comments. The comments by anonymous referees were substantially helpful for revising the article.

Funding

Financial support from the ERC Consolidator Grant (SNP 615882) is gratefully acknowledged (Otsu).

Notes

1 If ν is a smooth function of means, then ${\widehat{\nu }}^{*}$ is given by replacing the moments with the bootstrap counterparts. If $\widehat{\nu }$ is an M-estimator, we obtain ${\widehat{\nu }}^{*}$ through properly recentered estimating equations as in Shorack (1982) and Lahiri (1992).

2 The idea of recentering estimating equations is developed in Shorack (1982) and Lahiri (1992). It is interesting to see whether such recentering induces a desirable higher order property in our setup as in Lahiri (1992).

3 This follows from the Lipschitz property of the support function, |s(X, p) − s(X, q)| ⩽ ‖X‖_H‖p − q‖ a.s. for any $p,q\in {\mathbb{S}}^d$, which ensures that $\{s(X,p):p\in {\mathbb{S}}^d\}$ is μ-Donsker by a standard empirical process argument (e.g., van der Vaart 1998, Example 19.7).

4 For the identified set Θ₀ = {θ: E[m(θ)] ⩽ 0} defined by a finite number of moment inequalities, Chernozhukov, Kocatulum, and Menzel (2015) proposed a confidence region that is invariant to arbitrary one-to-one mappings of the form τ: Θ₀ → Ψ. However, their construction does not apply in general to our setup which is concerned with testing $\mathbb{E}\left[X_i\right]={\Theta }_0$ implying the continuum of moment inequalities. In contrast, invariance of K_n is restricted to particular transformations (i.e., multiplication of both {X_i}ⁿ_{i = 1} and Θ₀ by some nonsingular matrix independent of i).

5 The null for the opposite direction ${\mathrm{H}}_0:\mathbb{E}\left[X\right]\subseteq {\Theta }_0\left(\nu \right)$ can be treated analogously.

6 When ν is defined by a smooth function of means, it can be treated as in Owen (2001, sec. 3.4).

7 Chandrasekhar et al. (2012) extended this model further to allow for y_L and y_U to be nonparametrically estimable functions. Although it is beyond the scope of this article, it would be interesting to extend our empirical likelihood approach to such situations.

8 As expected, however, the marked empirical likelihood test is computationally more expensive than the Wald test. In particular, for sample size n = 1000, the marked empirical likelihood test with 399 bootstrap repetitions has an average run time of 5.7 s as compared to 0.6 s for the Wald test.