On a class of stopping times for M-estimators

For a given score function [psi] = [psi](x, [theta]), let [theta]n be Huber's M-estimator for an unknown population parameter [theta]. Under some mild smoothness assumptions it is known that n1/2([theta]n - [theta]) is asymptotically normal. In this paper the stopping times [tau]c(m) = inf{n >= m: n1/2 [theta]n - [theta] > c } associated with the sequence of confidence intervals for [theta] are investigated. A useful representation of M-estimators is derived, which is also appropriate for proving laws of the iterated logarithm and Donskertype invariance principles for ([pi]n)n.