Institute of Mathematical Statistics Collections

Consistent scoring functions for quantiles

Kyrill Grant and Tilmann Gneiting

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Abstract

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

Source
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

Dates
First available in Project Euclid: 8 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.imsc/1362751186

Digital Object Identifier
doi:10.1214/12-IMSCOLL912

Mathematical Reviews number (MathSciNet)
MR3202632

Zentralblatt MATH identifier
1327.62035

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

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

Rights
Copyright © 2010, Institute of Mathematical Statistics

Citation

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. https://projecteuclid.org/euclid.imsc/1362751186


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References

  • [1] Cervera, J. L. and Muñoz, J. (1996). Proper scoring rules for fractiles. In Bayesian Statistics 5 (Bernardo, J. M., Berger, J. O., Dawid, A. P. and Smith, A. F. M., eds.) 513–519. Oxford Univ. Press.
  • [2] Gneiting, T. (2011). Quantiles as optimal point forecasts. Int. J. Forecasting 27 197–207.
  • [3] Gneiting, T. (2011). Making and evaluating point forecasts. J. Amer. Statist. Assoc. 106 746–762.
  • [4] Gneiting, T. and Raftery, A. E. (2007). Strictly proper scoring rules, prediction, and estimation. J. Amer. Statist. Assoc. 102 359–378.
  • [5] Grant, K. (2011). Scoring functions for quantiles. Diploma thesis, Univ. of Heidelberg.
  • [6] Jose, V. R. R. and Winkler, R. L. (2009). Evaluating quantile assessments. Operations Res. 57 1287–1297.
  • [7] Koenker, R. (2005). Quantile Regression. Cambridge Univ. Press.
  • [8] Koltchinskii, V. I. (1997). $M$-estimation, convexity and quantiles. Ann. Statist. 25 435–477.
  • [9] Osband, K. H. (1985). Providing incentives for better cost forecasting. Ph.D. thesis, Univ. of California, Berkeley.
  • [10] Saerens, M. (2000). Building cost functions minimizing to some summary statistics. IEEE Trans. Neur. Netw. 11 1263–1271.
  • [11] Savage, L. J. (1971). Elicitation of personal probabilities and expectations. J. Amer. Statist. Assoc. 66 783–801.
  • [12] Thomson, W. (1979). Eliciting production possibilities from a well-informed manager. J. Econ. Theo. 20 360–380.
  • [13] Wellner, J. A. (2009). Statistical functionals and the delta method. Lecture notes, available online at http://www.stat.washington.edu/people/jaw/COURSES/580s/581/LECTNOTES/ch7.pdf.