  This paper highlights a tension between semiparametric efficiency and
bootstrap consistency in the context of a canonical semiparametric estimation
problem, namely the problem of estimating the average density. It is shown that
although simple plug-in estimators suffer from bias problems preventing them
from achieving semiparametric efficiency under minimal smoothness conditions,
the nonparametric bootstrap automatically corrects for this bias and that, as a
result, these seemingly inferior estimators achieve bootstrap consistency under
minimal smoothness conditions. In contrast, several "debiased" estimators that
achieve semiparametric efficiency under minimal smoothness conditions do not
achieve bootstrap consistency under those same conditions.
