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May, 1977 Asymptotic Relations of $M$-Estimates and $R$-Estimates in Linear Regression Model
Jana Jureckova
Ann. Statist. 5(3): 464-472 (May, 1977). DOI: 10.1214/aos/1176343843

Abstract

Let $\hat{\mathbf{\Delta}}_M$ be an $M$-estimator (maximum-likelihood type estimator) and $\hat{\mathbf{\Delta}}_R$ be an $R$-estimator (rank estimator) of the parameter $\mathbf{\Delta} = (\Delta_1,\cdots, \Delta_p)$ in the linear regression model $X_{Ni} = \sum^p_{j=1} \Delta_jc_{ji} + e_i, i = 1,\cdots, N$. The asymptotic distribution of $\hat\mathbf{\Delta}_M - \hat\mathbf{\Delta}_R$ is derived for $p$ fixed and $N \rightarrow \infty,$ under some assumptions on the design matrix, on the error distribution $F$ and on the functions generating the respective estimators. The result has several consequences which have an interest of their own; among others, it is shown that to any $M$-estimator corresponds an $R$-estimator such that the estimators asymptotically equivalent, and conversely. A special case when $\hat\mathbf{\Delta}_M$ is the maximum likelihood estimator and $\hat\mathbf{\Delta}_R$ the $R$-estimator, both asymptotically efficient for some distribution $G$, is also considered.

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Jana Jureckova. "Asymptotic Relations of $M$-Estimates and $R$-Estimates in Linear Regression Model." Ann. Statist. 5 (3) 464 - 472, May, 1977. https://doi.org/10.1214/aos/1176343843

Information

Published: May, 1977
First available in Project Euclid: 12 April 2007

zbMATH: 0365.62034
MathSciNet: MR433698
Digital Object Identifier: 10.1214/aos/1176343843

Subjects:
Primary: 62G05
Secondary: 62G35

Keywords: $M$-estimate , $R$-estimate , asymptotically normal distribution

Rights: Copyright © 1977 Institute of Mathematical Statistics

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Vol.5 • No. 3 • May, 1977
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