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 defi ning weighted empirical distributions. A Euclidean geometry is proposed giv ing such perturbations both size and direction. The notion of the salience of a perturbation is emphasized. This approach has several benefits. A gene ral probability case weight analysis results. Answers to a number of outsta nding questions follow directly. Rescaled versions of the three usual finit e 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) perturbat ions involved. Their essential unity, both within and between subsets, is e vident geometrically. Finally it is shown how a relaxation strategy, in whi ch a high dimensional (O(C-n(m))) discrete problem is replaced by a low dim ensional (O(n)) continuous problem, can combine with (convex) optimization results to deliver better performance in challenging multiple-case influenc e problems. Further developments are briefly indicated.