  We propose a structure of a semiparametric two-component mixture model when
one component is parametric and the other is defined through linear constraints
on its distribution function. Estimation of a two-component mixture model with
an unknown component is very difficult when no particular assumption is made on
the structure of the unknown component. A symmetry assumption was used in the
literature to simplify the estimation. Such method has the advantage of
producing consistent and asymptotically normal estimators, and identifiability
of the semiparametric mixture model becomes tractable. Still, existing methods
which estimate a semiparametric mixture model have their limits when the
parametric component has unknown parameters or the proportion of the parametric
part is either very high or very low. We propose in this paper a method to
incorporate a prior linear information about the distribution of the unknown
component in order to better estimate the model when existing estimation
methods fail. The new method is based on $\varphi-$divergences and has an
original form since the minimization is carried over both arguments of the
divergence. The resulting estimators are proved to be consistent and
asymptotically normal under standard assumptions. We show that using the
Pearson's $\chi^2$ divergence our algorithm has a linear complexity when the
constraints are moment-type. Simulations on univariate and multivariate
mixtures demonstrate the viability and the interest of our novel approach.
