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Biometrika 1988 75(4):761-771; doi:10.1093/biomet/75.4.761
© 1988 by Biometrika Trust
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M-quantiles

JENS BRECKLING and RAY CHAMBERS

Australian Bureau of Agricultural and Resource Economics Canberra City, ACT 2601, Australia

It is well known that a M-estimator of the centre of symmetry {Theta} of a symmetric distribution can be defined in terms of either a continuous symmetric loss function {rho} or the associated influence function{psi}. This estimator is robust if {psi} is bounded. In this paper, we develop a generalization of the M-concept to estimation of a quantile analogue of {Theta}, called a M-quantile, by introducing a particular kind of asymmetry into {psi}. A natural consequence is that the M- quantile parameter then bears the same relationship to {Theta} as the ordinary quantile bears to the median. The extension of this idea to the case of multivariate data is considered in some detail. Two applications to the analysis of agricultural survey data are given.

Key Words: Influence curve • M-estimation • Multivariate data • Quantile • Regression • Robust estimation


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