Finite sample stability properties of the least median of squares estimator

This article examines the stability properties of the least median of squares (LMS) estimate. Attention is focussed on LMS since it is arguably the most widely used high breakdown regression estimate. The differing roles of breakdown point and influence function in producing estimates which are stable to changes in the data are discussed. Simulations and real examples are used to illustrate the extent to which the LMS estimate can change when small changes are made to centrally located data. It is shown that this instability is a consequence of the fact that the LMS estimate has an influence function which is unbounded to the effects of centrally located x's and is not merely a consequence of the exact fit property and the curse of dimensionality, as argued by Rousseeuw (1994).