On the estimation of cross-information quantities in rank-based inference

Abstract

Rank-based inference and, in particular, R-estimation, is a red thread running through Jana Jurečková’s entire scientific career, starting with her dissertation in 1967, where she laid the foundations of an extension to linear regression of the R-estimation methods that had recently been proposed by Hodges and Lehmann [13]. Cross-information quantities in that context play an essential role. In location/regression problems, these quantities take the form ∫₀¹φ(u)φ_g(u) du where φ is a score function and φ_g(u):=g'(G⁻¹(u))/g(G⁻¹(u)) is the log-derivative of the unknown actual underlying density g computed at the quantile G⁻¹(u); in other models, they involve more general scores. Such quantities appear in the local powers of rank tests and the asymptotic variance of R-estimators. Estimating them consistently is a delicate problem that has been extensively considered in the literature. We provide here a new, flexible, and very general method for that problem, which furthermore applies well beyond the traditional case of regression models.