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

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