Nonparametric Robust Comparison of Solutions under Input Uncertainty

TL;DR

This paper introduces a nonparametric method, NIOU-C, for reliably identifying optimal solutions under input uncertainty without additional data collection, using empirical likelihood and ambiguity sets.

Contribution

It develops a novel nonparametric procedure, NIOU-C, for constructing confidence sets that include the optimal solution under input uncertainty, with an extension NIOU-C:E to reduce conservatism.

Findings

NIOU-C produces smaller confidence sets than parametric methods.

NIOU-C reliably includes the true optimum more often.

Sample size requirements for asymptotic validity are characterized.

Abstract

We study ranking and selection under input uncertainty in settings where additional data cannot be collected. We propose the Nonparametric Input-Output Uncertainty Comparisons (NIOU-C) procedure to construct a confidence set that includes the optimal solution with a user-specified probability. We construct an ambiguity set of input distributions using empirical likelihood and approximate the mean performance of each solution using a linear functional representation of the input distributions. By solving optimization problems evaluating worst-case pairwise mean differences within the ambiguity set, we build a confidence set of solutions indistinguishable from the optimum. We characterize sample size requirements for NIOU-C to achieve the asymptotic validity under mild conditions. Moreover, we propose an extension to NIOU-C, NIOU-C:E, that mitigates conservatism and yields a smaller confidence set. In numerical experiments, NIOU-C provides a smaller confidence set that includes the optimum more frequently than a parametric procedure that takes advantage of the parametric distribution families.