  Statistical models of unobserved heterogeneity are typically formalized as
mixtures of simple parametric models and interest naturally focuses on testing
for homogeneity versus general mixture alternatives. Many tests of this type
can be interpreted as $C(\alpha)$ tests, as in Neyman (1959), and shown to be
locally, asymptotically optimal. These $C(\alpha)$ tests will be contrasted
with a new approach to likelihood ratio testing for general mixture models. The
latter tests are based on estimation of general nonparametric mixing
distribution with the Kiefer and Wolfowitz (1956) maximum likelihood estimator.
Recent developments in convex optimization have dramatically improved upon
earlier EM methods for computation of these estimators, and recent results on
the large sample behavior of likelihood ratios involving such estimators yield
a tractable form of asymptotic inference. Improvement in computation efficiency
also facilitates the use of a bootstrap methods to determine critical values
that are shown to work better than the asymptotic critical values in finite
samples. Consistency of the bootstrap procedure is also formally established.
We compare performance of the two approaches identifying circumstances in which
each is preferred.
