We suggest methods for tilting a likelihood so as to enhance the robustness
of maximum likelihood procedures. From the viewpoint of computation, tilti
ng amounts to choosing unequal weights for the score function in such a way
as to maximise likelihood subject to moving a given distance from equally
weighted scores. Empirical methods, based on standard parametric Q-Q plots,
are used to determine the appropriate amount of tilting. Distance may be m
easured in a variety of ways, and we devote particular attention to power-d
ivergence approaches. In this context, one of the two Kullback-Leibler dist
ance measures is shown to be advantageous.