  A new estimation method for the two-component mixture model introduced in
\cite{Van13} is proposed. This model consists of a two-component mixture of
linear regressions in which one component is entirely known while the
proportion, the slope, the intercept and the error distribution of the other
component are unknown. In spite of good performance for datasets of reasonable
size, the method proposed in \cite{Van13} suffers from a serious drawback when
the sample size becomes large as it is based on the optimization of a contrast
function whose pointwise computation requires O(n^2) operations. The range of
applicability of the method derived in this work is substantially larger as it
relies on a method-of-moments estimator free of tuning parameters whose
computation requires O(n) operations. From a theoretical perspective, the
asymptotic normality of both the estimator of the Euclidean parameter vector
and of the semiparametric estimator of the c.d.f.\ of the error is proved under
weak conditions not involving zero-symmetry assumptions. In addition, an
approximate confidence band for the c.d.f.\ of the error can be computed using
a weighted bootstrap whose asymptotic validity is proved. The finite-sample
performance of the resulting estimation procedure is studied under various
scenarios through Monte Carlo experiments. The proposed method is illustrated
on three real datasets of size $n=150$, 51 and 176,343, respectively. Two
extensions of the considered model are discussed in the final section: a model
with an additional scale parameter for the first component, and a model with
more than one explanatory variable.
