Abstract
In regression scenarios there is a growing demand for information on the conditional distribution of the response beyond the mean. In this scenario quantile regression is an established method of tail analysis. It is well understood in terms of asymptotic properties and estimation quality. Another way to look at the tail of a distribution is via expectiles. They provide a valuable alternative since they come with a combination of preferable attributes. The easy weighted least squares estimation of expectiles and the quadratic penalties often used in flexible regression models are natural partners. Also, in a similar way as quantiles can be seen as a generalisation of median regression, expectiles offer a generalisation of mean regression. In addition to regression estimates, confidence intervals are essential for interpretational purposes and to assess the variability of the estimate, but there is a lack of knowledge regarding the asymptotic properties of a semiparametric expectile regression estimate. Therefore confidence intervals for expectiles based on an asymptotic normal distribution are introduced. Their properties are investigated by a simulation study and compared to a boostrap-based gold standard method. Finally the introduced confidence intervals help to evaluate a geoadditive expectile regression model on childhood malnutrition data from India.
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References
Breckling, J., Chambers, R.: M-quantiles. Biometrika 75, 761–771 (1988)
Buchinsky, M.: Recent advances in quantile regression models: a practical guideline for empirical research. J. Hum. Resour. 33, 88–126 (1998)
Efron, B.: Regression percentiles using asymmetric squared error loss. Stat. Sin. 1, 93–125 (1991)
Efron, B., Tibshirani, R.J.: An Introduction to the Bootstrap, 1st edn. Chapman and Hall, New York (1993)
Eilers, P.H.C., Marx, B.D.: Flexible smoothing with B-splines and penalties. Stat. Sci. 11, 89–121 (1996)
Fahrmeir, L., Kneib, T., Lang, S.: Penalized structured additive regression: a Bayesian perspective. Stat. Sin. 14, 731–761 (2004)
Fenske, N., Kneib, T., Hothorn, T.: Identifying risk factors for severe childhood malnutrition by boosting additive quantile regression. J. Am. Stat. Assoc. 106(494), 494–510 (2011)
Jones, M.: Expectiles and m-quantiles are quantiles. Stat. Probab. Lett. 20(2), 149–153 (1994)
Kammann, E.E., Wand, M.P.: Geoadditive models. Appl. Stat. 52, 1–18 (2003)
Kocherginsky, M., He, X., Mu, Y.: Practical confidence intervals for regression quantiles. J. Comput. Graph. Stat. 14, 41–55 (2005)
Koenker, R.: Quantile Regression. Cambridge University Press, New York (2005)
Koenker, R., Bassett, G.: Regression quantiles. Econometrica 46, 33–50 (1978)
Meinshausen, N., Bühlmann, P.: Stability selection. J. R. Stat. Soc., Ser. B 72(4) (2010, in press), with discussion. doi:10.1111/j.1467-9868.2010.00740.x
Newey, W.K., Powell, J.L.: Asymmetric least squares estimation and testing. Econometrica 55, 819–847 (1987)
R Development Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria (2010). http://www.R-project.org, ISBN 3-900051-07-0
Rigby, R., Stasinopoulos, D.: Generalized additive models for location, scale and shape. Appl. Stat. 54, 507–554 (2005)
Rue, H., Held, L.: Gaussian Markov Random Fields. Chapman & Hall/CRC, Boca Raton (2005)
Schall, R.: Estimation in generalized linear models with random effects. Biometrika 78, 719–727 (1991)
Schnabel, S., Eilers, P.: Optimal expectile smoothing. Comput. Stat. Data Anal. 53, 4168–4177 (2009)
Sobotka, F., Kneib, T.: Geoadditive expectile regression. Comput. Stat. Data Anal. (2010). doi:10.1016/j.csda.2010.11.015
Sobotka, F., Schnabel, S., Schulze Waltrup, L.: expectreg: Expectile and Quantile Regression. http://CRAN.R-project.org/package=expectreg, r package version 0.26 (2011)
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Sobotka, F., Kauermann, G., Schulze Waltrup, L. et al. On confidence intervals for semiparametric expectile regression. Stat Comput 23, 135–148 (2013). https://doi.org/10.1007/s11222-011-9297-1
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DOI: https://doi.org/10.1007/s11222-011-9297-1