  Suppose that univariate data are drawn from a mixture of two distributions
that are equal up to a shift parameter. Such a model is known to be
nonidentifiable from a nonparametric viewpoint. However, if we assume that the
unknown mixed distribution is symmetric, we obtain the identifiability of this
model, which is then defined by four unknown parameters: the mixing proportion,
two location parameters and the cumulative distribution function of the
symmetric mixed distribution. We propose estimators for these four parameters
when no training data is available. Our estimators are shown to be strongly
consistent under mild regularity assumptions and their convergence rates are
studied. Their finite-sample properties are illustrated by a Monte Carlo study
and our method is applied to real data.
