  Patra and Sen (2016) consider a two-component mixture model, where one
component plays the role of background while the other plays the role of
signal, and propose to estimate the background component by simply "maximizing"
its weight. While in their work the background component is a completely known
distribution, we extend their approach here to three emblematic settings: when
the background distribution is symmetric; when it is monotonic; and when it is
log-concave. In each setting, we derive estimators for the background
component, establish consistency, and provide a confidence band. While the
estimation of a background component is straightforward when it is taken to be
symmetric or monotonic, when it is log-concave its estimation requires the
computation of a largest concave minorant, which we implement using sequential
quadratic programming. Compared to existing methods, our method has the
advantage of requiring much less prior knowledge on the background component,
and is thus less prone to model misspecification. We illustrate this
methodology on a number of synthetic and real datasets.
