Rendering parametric procedures more robust by empirically tilting the model

We suggest methods for tilting a likelihood so as to enhance the robustness of maximum likelihood procedures. From the viewpoint of computation, tilting 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 measured in a variety of ways, and we devote particular attention to power-divergence approaches. In this context, one of the two Kullback-Leibler distance measures is shown to be advantageous.