Partial breakdown in two-factor models

The usual (global) breakdown point describes the worst effect that a given number of gross errors can have. In a two-way layout, without interaction, one is frustrated by the small number of gross errors such a design can tolerate. However, neither the whole fit nor all parameter estimates need to be affected by such a breakdown. An example from molecular spectroscopy serves to illustrate such partial breakdown in a large, “sparse” two-factor model. Because the global finite sample breakdown point is zero for all usual estimators in this example, this concept does not make sense in such problems. The more appropriate concept of partial breakdown point is discussed in this paper. It also provides a crude quantification of the robustness properties of an estimator, yet for any linear combination of the estimated parameters. The maximum number of gross errors to which the linear combination of the estimated parameters can resist is related to the minimum number of observations that must be omitted to make the linear function a non-estimable function. In the example, we are mainly interested in differences of parameters. Then the maximal partial breakdown point for regression equivariant estimators is one half, and Huber-type regression M-estimators with bounded ψ-function reach this limit.