Asymptotic Relations of $M$-Estimates and $R$-Estimates in Linear Regression Model

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.