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December 1996 A general Bahadur representation of M-estimators and its application to linear regression with nonstochastic designs
Xuming He, Qi-Man Shao
Ann. Statist. 24(6): 2608-2630 (December 1996). DOI: 10.1214/aos/1032181172

Abstract

We obtain strong Bahadur representations for a general class of M-estimators that satisfies $\Sigma_i \psi (x_i, \theta) = o(\delta_n)$, where the $x_i$'s are independent but not necessarily identically distributed random variables. The results apply readily to M-estimators of regression with nonstochastic designs. More specifically, we consider the minimum $L_p$ distance estimators, bounded influence GM-estimators and regression quantiles. Under appropriate design conditions, the error ratesobtained for the first-order approximations are sharp in these cases. We also provide weaker and more easily verifiable conditions that suffice for an error rate that is suboptimal but strong enough for deriving the asymptotic distribution of M-estimators in a wide variety of problems.

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Xuming He. Qi-Man Shao. "A general Bahadur representation of M-estimators and its application to linear regression with nonstochastic designs." Ann. Statist. 24 (6) 2608 - 2630, December 1996. https://doi.org/10.1214/aos/1032181172

Information

Published: December 1996
First available in Project Euclid: 16 September 2002

zbMATH: 0867.62012
MathSciNet: MR1425971
Digital Object Identifier: 10.1214/aos/1032181172

Subjects:
Primary: 62F12, 62J05
Secondary: 60F15, 62F35

Rights: Copyright © 1996 Institute of Mathematical Statistics

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Vol.24 • No. 6 • December 1996
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