Silence Diagnostics: The Influence Curve Revisited

The influence curve (Hampel, 1974), latterly known as the influence function, lies at the heart of an established approach to robust statistics. Finite sample versions of it also form the foundation of that part of diagnostics known as influence analysis. In their booklength exposition, Hampel at al. (1986), while providing a formal mathematical derivation, emphasise that:

“The importance of the influence function lies in its heuristic interpretation: it describes the effect of an infinitesimal contamination at the point $x$ on the estimate, standardised by the amount of contamination.”

The present talk explores the extent to which this heuristic interpretation can itself be formalised. It is based on developments of the concept of salience and of the perturbation geometry introduced in Critchely et al. (2001).

References

• Critchley, F., Atkinson, R.A., Lu, G. and Biazi, E. (2001). Influence analysis based on the case sensitivity function. J. Royal Statist. Soc., B. 63(2), 307-323.
• Hampel, F.R. (1974). The influence curve and its role in robust estimation. J. Am. Satist. Soc., 69, 383-393.
• Hampel, F.R., Ronchetti, E.M. Rousseeuw, P.J. and Stahel, W.A. (1986). Robust Statistics: The Approach Based on Influence Functions. Wiley.