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M-quantiles. (English) Zbl 0653.62024
Summary: It is well known that an 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 an 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.

62F35 Robustness and adaptive procedures (parametric inference)
62J05 Linear regression; mixed models
62P10 Applications of statistics to biology and medical sciences; meta analysis
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