INFLUENCE FUNCTIONS FOR ITERATIVELY DEFINED STATISTICS

Hampel's influence function and its finite-sample counterparts are the basis for a number of diagnostic statistics. These diagnostics can be expensive to compute in the natural way when the estimation calculati ons are iterative, as they frequently are when maximum likelihood or r obust methods are used. We show how the influence function can be calc ulated in these situations by implicit differentiation of the fixed-po int equation satisfied by the limit of the iterative process. We consi der in particular the cases of Newton's method and iteratively reweigh ted least squares where interesting analytic results are available. As an application we consider the generalization of Pregibon's (1981) lo gistic regression diagnostics to cover generalized linear models with non-canonical link functions such as probit regression.