Consistent scoring functions for quantiles. (English) Zbl 1327.62035

Grant, Kyrill; Gneiting, Tilmann

Banerjee, M. (ed.) et al., From probability to statistics and back: high-dimensional models and processes. A Festschrift in honor of Jon A. Wellner. Including papers from the conference, Seattle, WA, USA, July 28–31, 2010. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 978-0-940600-83-6). Institute of Mathematical Statistics Collections 9, 163-173 (2013).

Summary: A scoring function is consistent for the \(\alpha\)-quantile functional if, and only if, it is generalized piecewise linear (GPL) of order \(\alpha\), up to equivalence. Expressed differently, loss functions that yield quantiles as Bayes rules are GPL functions. We review and discuss this basic decision-theoretic result with focus on Thomson’s pioneering characterization.
For the entire collection see [Zbl 1319.62002].

MSC:

62C05	General considerations in statistical decision theory
91B06	Decision theory

Keywords:

Bayes rule; consistent scoring function; fractile; optimal point forecast; proper scoring rule; quantile