Abstract
Recently, quantiles and expectiles of a regression function have been investigated by several authors. In this work, we give a sufficient condition under which a quantile and an expectile coincide. We extend some classical results known for mean, median and symmetry to expectiles, quantiles and weighted-symmetry. We also study split-models and sample estimators of expectiles.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abdous, B. (1992). Kernel estimators of regression expectiles, Tech. Report, Université du Québec à Trois-Rivières.
Aigner, D., Amemiya, T. and Poirier, D. (1976). On the estimation of production frontiers: maximum likelihood estimation of the parameters of a discontinuous density function,Internat. Econom. Rev.,17, 372–396.
Aki, S. (1987). On nonparametric tests for symmetry,Ann. Inst. Statist. Math.,39, 457–472.
Breckling, J. and Chambers, R. (1988).M-quantiles,Biometrika,75, 761–771.
Efron, B. (1991). Regression percentiles using asymmetric squared error loss,Statistica Sinica,1, 93–125.
Fechner, G. Th. (1897).Kollektivmasslehre, Leipzig, Engelman.
Gibbons, J. F. and Mylroie, S. (1973). Estimation of impurity profiles in ion-implanted amorphous targets using joined half-Gaussian distributions,Applied Physical Letters,22, 568–569.
Gibbons, J. F., Johnson, W. S. and Mylroie, S. (1975).Projected Range Statistics, 2nd ed., Wiley, New York.
Gupta, M. K. (1967). An asymptotically nonparametric test of symmetry,Ann. Math. Statist., 849–866.
Hampel, F. R. (1974). The influence curve and its role in robust estimation,J. Amer. Statist. Assoc.,69, 383–393.
John, S. (1982). The three-parameter two-piece normal family of distributions and its fitting,Comm. Statist. Theory Methods,11, 879–885.
Kimber, A. C. (1985). Methods for the two-piece normal distribution,Comm. Statist. Theory Methods,14, 235–245.
Koenker, R. and Basset, G. (1978). Regression quantiles,Econometrica,46, 33–50.
Lefrançois, P. (1989). Allowing for asymmetry in forecast errors: Results from a Monte-Carlo study,International Journal of Forcasting,5, 99–110.
Lehmann, E. L. (1953). The power of rank tests,Ann. Math. Statist.,24, 23–43.
Lehmann, E. L. (1983).Theory of Point Estimation, Wiley, New York.
Nabeya, S. (1987) On Aki's nonparametric test for symmetry,Ann. Inst. Statist. Math.,39, 473–482.
Newey, W. K. and Powell, J. L. (1987). Asymmetric least squares estimation and testing,Econometrica,55, 819–847.
Parent, E. A. (1965). Sequential ranking procedures, Tech. Report, No. 80, Department of Statistics, Stanford University, California.
Rosenberger, J. L. and Gasko, M. (1983). Comparing location estimators: trimmed means, medians, and trimean,Understanding Robust and Exploratory Data Analysis (eds. D. C. Hoaglin, F. Mosteller and J. W. Tukey), 297–338, Wiley, New York.
Runnenburg, J. Th. (1978). Mean, median, mode,Statistica Neerlandica,32, 73–79.
Serfling, R. J. (1980).Approximation Theorems of Mathematical Statistics, Wiley, New York.
van Zwet, W. R. (1979). Mean, median, mode II,Statistica Neerlandica,33, 1–5.
Wang, Y. (1992). Smoothing splines for non-parametric regression percentiles, Tech. Report, No. 9207, Department of Statistics, University of Toronto.
Wolfe, D. A. (1974). A characterization of population weighted-symmetry and related results,J. Amer. Statist. Assoc.,69 819–822.
Author information
Authors and Affiliations
Additional information
Work supported by the Natural Science and Engineering Council of Canada and by the Université du Québec à Trois-Rivières.
About this article
Cite this article
Abdous, B., Remillard, B. Relating quantiles and expectiles under weighted-symmetry. Ann Inst Stat Math 47, 371–384 (1995). https://doi.org/10.1007/BF00773468
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00773468