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.