  We consider estimation of a one-dimensional location parameter by means of
M-estimators S_n with monotone influence curve psi. For growing sample size n,
on suitably thinned out convex contamination ball BQ_n of shrinking radius
r/sqrt(n) about the ideal distribution, we obtain an expansion of the
asymptotic maximal mean squared error MSE of form r^2 sup psi^2 + E_{id} psi^2
+ r/sqrt(n) A_1 + 1/n A_2 + o(1/n), where A_1, A_2 are constants depending on
psi and r. Hence S_n not only is uniformly (square) integrable in n (in the
ideal model) but also on BQ_n, which is not self-evident. For this result, the
thinning of the neighborhoods, by a breakdown-driven, sample-wise restriction,
is crucial, but exponentially negligible. Moreover, our results essentially
characterize contaminations generating maximal MSE up to o(1/n).
  Our results are confirmed empirically by simulations as well as numerical
evaluations of the risk.
