×

zbMATH — the first resource for mathematics

On \(L\)-estimators viewed as \(M\)-estimators. (English) Zbl 0828.62022
Summary: Arithmetical mean and median usually serve as basic examples of \(M\)- estimators. Both of them are \(L\)-estimators. Thus there is a natural question whether there are some other \(L\)-estimators which are \(M\)- estimators as well. We shall show that, with rare exceptions, this is not the case. More precisely, we shall show that the arithmetical mean and empirical quantiles are the only \(L\)-estimators with nonnegative coefficients having a nontrivial \(\psi\)-function.

MSC:
62F10 Point estimation
62G05 Nonparametric estimation
PDF BibTeX XML Cite
Full Text: EuDML Link
References:
[1] J. Aczél: Lectures on Functional Equations and their Applications. Academic Press, New York–London 1966. · Zbl 0139.09301
[2] F. R. Hampel E. M. Ronchetti P. J. Rousseeuw, and W. A. Stahel: Robust Statistics. The Approach Based on Influence Functions. John Wiley & Sons, New York–Chichester–Brisbane–Toronto–Singapore 1986.
[3] P. J. Huber: Robust Statistics. John Wiley & Sons, New York–Chichester–Brisbane–Toronto 1981. · Zbl 0536.62025
[4] P. Kovanic: Gnostical theory of small samples of real data. Problems Control Inform. Theory 13 (1984), 5, 303-319. · Zbl 0583.94008
[5] E. L. Lehman: Theory of point estimation. J. Wiley, New York–Chichester– Brisbane–Toronto–Singapore 1983.
[6] J. Novovičová: M-estimators and gnostical estimators for identification of a regression model. Automatica 26 (1990), 3, 607-610. · Zbl 0713.93061
[7] I. Vajda: Theory of Statistical Inference and Information. Kluwer Academic Publishers, Dordrecht–Boston 1989. · Zbl 0711.62002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.