  In a multiple testing context, we consider a semiparametric mixture model
with two components where one component is known and corresponds to the
distribution of $p$-values under the null hypothesis and the other component
$f$ is nonparametric and stands for the distribution under the alternative
hypothesis. Motivated by the issue of local false discovery rate estimation, we
focus here on the estimation of the nonparametric unknown component $f$ in the
mixture, relying on a preliminary estimator of the unknown proportion $\theta$
of true null hypotheses. We propose and study the asymptotic properties of two
different estimators for this unknown component. The first estimator is a
randomly weighted kernel estimator. We establish an upper bound for its
pointwise quadratic risk, exhibiting the classical nonparametric rate of
convergence over a class of H\"older densities. To our knowledge, this is the
first result establishing convergence as well as corresponding rate for the
estimation of the unknown component in this nonparametric mixture. The second
estimator is a maximum smoothed likelihood estimator. It is computed through an
iterative algorithm, for which we establish a descent property. In addition,
these estimators are used in a multiple testing procedure in order to estimate
the local false discovery rate. Their respective performances are then compared
on synthetic data.
