Influence analysis based on the case sensitivity function

The case sensitivity function approach to influence analysis is introduced as a natural smooth extension of influence curve methodology in which both the insights of geometry and the power of (convex) analysis are available. In it, perturbation is defined as movement between probability vectors defining weighted empirical distributions. A Euclidean geometry is proposed giving such perturbations both size and direction. The notion of the salience of a perturbation is emphasized. This approach has several benefits. A general probability case weight analysis results. Answers to a number of outstanding questions follow directly. Rescaled versions of the three usual finite sample influence curve measures-seen now to be required for comparability across different-sized subsets of cases - are readily available. These new diagnostics directly measure the salience of the (infinitesimal) perturbations involved. Their essential unity, both within and between subsets, is evident geometrically. Finally it is shown how a relaxation strategy, in which a high dimensional (O(C-n(m))) discrete problem is replaced by a low dimensional (O(n)) continuous problem, can combine with (convex) optimization results to deliver better performance in challenging multiple-case influence problems. Further developments are briefly indicated.