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Consistent scoring functions for quantiles

Kyrill Grant and Tilmann Gneiting

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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.

Chapter information

Banerjee, M., Bunea, F., Huang, J., Koltchinskii, V., and Maathuis, M. H., eds., From Probability to Statistics and Back: High-Dimensional Models and Processes -- A Festschrift in Honor of Jon A. Wellner, (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2013) , 163-173

First available in Project Euclid: 8 March 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62C05: General considerations
Secondary: 91B06: Decision theory [See also 62Cxx, 90B50, 91A35]

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

Copyright © 2010, Institute of Mathematical Statistics


Grant, Kyrill; Gneiting, Tilmann. Consistent scoring functions for quantiles. From Probability to Statistics and Back: High-Dimensional Models and Processes -- A Festschrift in Honor of Jon A. Wellner, 163--173, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2013. doi:10.1214/12-IMSCOLL912.

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