  We consider in this paper the semiparametric mixture of two distributions
equal up to a shift parameter. The model is said to be semiparametric in the
sense that the mixed distribution is not supposed to belong to a parametric
family. In order to insure the identifiability of the model it is assumed that
the mixed distribution is symmetric, the model being then defined by the mixing
proportion, two location parameters, and the probability density function of
the mixed distribution. We propose a new class of M-estimators of these
parameters based on a Fourier approach, and prove that they are square root
consistent under mild regularity conditions. Their finite-sample properties are
illustrated by a Monte Carlo study and a benchmark real dataset is also studied
with our method.
