  We introduce a new method of estimation of parameters in semiparametric and
nonparametric models. The method is based on estimating equations that are
$U$-statistics in the observations. The $U$-statistics are based on higher
order influence functions that extend ordinary linear influence functions of
the parameter of interest, and represent higher derivatives of this parameter.
For parameters for which the representation cannot be perfect the method leads
to a bias-variance trade-off, and results in estimators that converge at a
slower than $\sqrt n$-rate. In a number of examples the resulting rate can be
shown to be optimal. We are particularly interested in estimating parameters in
models with a nuisance parameter of high dimension or low regularity, where the
parameter of interest cannot be estimated at $\sqrt n$-rate, but we also
consider efficient $\sqrt n$-estimation using novel nonlinear estimators. The
general approach is applied in detail to the example of estimating a mean
response when the response is not always observed.
