Positive-breakdown regression by minimizing nested scale estimators

In this article we present a new class of robust regression estimators. Our main example will be called the least trimmed median estimator (LTM). It is based on the minimization of the objective function 1/h(p) Sigma(k=1)(hp) {median(j) \r(i)(beta) - r(j)(beta)\}((k)) where h(p) = [1/2(n + p + 1)] and the subscript (k) indicates the kth order statistic. It can be seen as an alternative to the least median of squares (LMS) and the least trimmed squares (LTS) estimators, which correspond to minimizing the objective functions \r\((hp)) and Sigma(k=1)(hp)r((k))(2). An important advantage of the LTM is that it is not geared towards symmetric error distributions, which makes it more generally applicable. We will see that the LTM has the same breakdown point as the LMS and the LTS, but that its gaussian efficiency is higher. We will also show that the LTM has a much better bias curve than the LTS, and that its computation is virtually the same. The LTM is illustrated on a real data set about concentrations of plutonium isotopes.