Summary: We consider the multiple linear regression model \(y_ i= {\mathbf x}_ i'\beta+ \varepsilon_ i\), \(i=1,2,\dots,n\), with random carriers and focus on the estimation of \(\beta\). This article’s main contribution is to present an estimator that is affine, regression, and scale equivariant; has both a high breakdown point and a bounded influence function; and has an asymptotic efficiency greater than .95 versus least squares under Gaussian errors. We give conditions under which the estimator – a one-step general \(M\) estimator that uses Schweppe weights and is based on a high breakdown initial estimator – satisfies these properties.
The major conditions necessary for the estimator are (a) it must be based on a \(\sqrt n\)-consistent initial estimator with a \(50\%\) breakdown point, and (b) it must be based on a psi function that is odd, bounded, and strictly increasing. The advantage of this estimator over previous approaches is that it does not downweight high leverage points without first considering how they fit the bulk of the data. Methods of computing diagnostics and constructing Wald-type tests about \(\beta\) are given. We illustrate the features of the estimator on a data set with two regressors, showing how a good leverage point is not downweighted, whereas a bad leverage point is downweighted.