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On the Lp-quantiles for the Student t distribution

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  • Bernardi, Mauro
  • Bignozzi, Valeria
  • Petrella, Lea

Abstract

Lp-quantiles are a class of generalised quantiles defined as the minimisers of an expected asymmetric power function. For p=1 and p=2 they correspond respectively to the quantiles and the expectiles. In this contribution we show that for the class of Student t distributions with p degrees of freedom, the Lp-quantile and the quantile coincide for any confidence level τ∈(0,1). The proof involves concepts from combinatorial analysis as well as a recursive formula for the truncated moments of the Student t distribution. This work extends the contribution of Koenker (1993) that shows a similar result for the expectiles.

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  • Bernardi, Mauro & Bignozzi, Valeria & Petrella, Lea, 2017. "On the Lp-quantiles for the Student t distribution," Statistics & Probability Letters, Elsevier, vol. 128(C), pages 77-83.
    • Handle: RePEc:eee:stapro:v:128:y:2017:i:c:p:77-83
    DOI: 10.1016/j.spl.2017.04.017
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    References listed on IDEAS

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    1. Belkacem Abdous & Bruno Remillard, 1995. "Relating quantiles and expectiles under weighted-symmetry," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 47(2), pages 371-384, June.
    2. Mauro Bernardi & Ghislaine Gayraud & Lea Petrella, 2013. "Bayesian inference for CoVaR," Papers 1306.2834, arXiv.org, revised Nov 2013.
    3. Yu, Keming & Moyeed, Rana A., 2001. "Bayesian quantile regression," Statistics & Probability Letters, Elsevier, vol. 54(4), pages 437-447, October.
    4. Gneiting, Tilmann, 2011. "Making and Evaluating Point Forecasts," Journal of the American Statistical Association, American Statistical Association, vol. 106(494), pages 746-762.
    5. Chen, Zehua, 1996. "Conditional Lp-quantiles and their application to the testing of symmetry in non-parametric regression," Statistics & Probability Letters, Elsevier, vol. 29(2), pages 107-115, August.
    6. Ray Chambers & Nikos Tzavidis, 2006. "M-quantile models for small area estimation," Biometrika, Biometrika Trust, vol. 93(2), pages 255-268, June.
    7. Bellini, Fabio & Klar, Bernhard & Müller, Alfred & Rosazza Gianin, Emanuela, 2014. "Generalized quantiles as risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 54(C), pages 41-48.
    8. Koenker, Roger W & Bassett, Gilbert, Jr, 1978. "Regression Quantiles," Econometrica, Econometric Society, vol. 46(1), pages 33-50, January.
    9. Bellini, Fabio, 2012. "Isotonicity properties of generalized quantiles," Statistics & Probability Letters, Elsevier, vol. 82(11), pages 2017-2024.
    10. Taddy, Matthew A. & Kottas, Athanasios, 2010. "A Bayesian Nonparametric Approach to Inference for Quantile Regression," Journal of Business & Economic Statistics, American Statistical Association, vol. 28(3), pages 357-369.
    11. Koenker, Roger, 1993. "When are Expectiles Percentiles?," Econometric Theory, Cambridge University Press, vol. 9(03), pages 526-527, June.
    12. Newey, Whitney K & Powell, James L, 1987. "Asymmetric Least Squares Estimation and Testing," Econometrica, Econometric Society, vol. 55(4), pages 819-847, July.
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    Cited by:

    1. Bernardi, Mauro & Bottone, Marco & Petrella, Lea, 2018. "Bayesian quantile regression using the skew exponential power distribution," Computational Statistics & Data Analysis, Elsevier, vol. 126(C), pages 92-111.
    2. Valeria Bignozzi & Luca Merlo & Lea Petrella, 2022. "Inter-order relations between moments of a Student $t$ distribution, with an application to $L_p$-quantiles," Papers 2209.12855, arXiv.org.

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