Enno Mammen, University of Heidelberg


Nonparametric smoothing methods for a class of nonstandard
curve estimation problems.


We discuss a class of nonparametric curve estimation problems
where a curve $m$ is defined as solution of a linear integral
equation
$$ m = m^* +  {\cal H} m,$$
where $\cal H$ is a linear integral operator and $m^*$ is another
curve. We suppose that both, $\cal H$ and $m^*$, are unknown,
but can be estimated by standard smoothing methods. Such
estimation problems include models that have been considered
from another point of view, e.g. additive models, panels of time series
and nonparametric regression with correlated  errors. New examples
include semiparametric GARCH models and nonparametric estimation
of yield curves.  The new framework motivates a new class of estimates
that are defined by plugging classical smoothing estimates
$\widehat {\cal H}$ and $\widehat m^*$ into the integral equation and
by solving
$$\widehat m =\widehat m^* + \widehat {\cal H} \widehat m.$$
Using the analytic  understanding of the pilot estimates it is possible
to develop a complete mathematical asymptotic theory for the estimate
$\widehat m$. The talk will discuss usefulness of this model class
and will outline the asymptotic theory.