  In this paper, different strands of literature are combined in order to
obtain algorithms for semi-parametric estimation of discrete choice models that
include the modelling of unobserved heterogeneity by using mixing distributions
for the parameters defining the preferences. The models use the theory on
non-parametric maximum likelihood estimation (NP-MLE) that has been developed
for general mixing models. The expectation-maximization (EM) techniques used in
the NP-MLE literature are combined with strategies for choosing appropriate
approximating models using adaptive grid techniques. \\ Jointly this leads to
techniques for specification and estimation that can be used to obtain a
consistent specification of the mixing distribution. Additionally, also
algorithms for the estimation are developed that help to decrease problems due
to the curse of dimensionality. \\ The proposed algorithms are demonstrated in
a small scale simulation study to be useful for the specification and
estimation of mixture models in the discrete choice context providing some
information on the specification of the mixing distribution. The simulations
document that some aspects of the mixing distribution such as the expectation
can be estimated reliably. They also demonstrate, however, that typically
different approximations to the mixing distribution lead to similar values of
the likelihood and hence are hard to discriminate. Therefore it does not appear
to be possible to reliably infer the most appropriate parametric form for the
estimated mixing distribution.
