  A unified framework is proposed for tests of unobserved heterogeneity in
parametric statistic models based on Neyman's $C(\alpha)$ approach. Such tests
are irregular in the sense that the first order derivative of the log
likelihood with respect to the heterogeneity parameter is identically zero, and
consequently the conventional Fisher information about the parameter is zero.
Nevertheless, local asymptotic optimality of the $C(\alpha)$ tests can be
established via LeCam's differentiability in quadratic mean and the limit
experiment approach. This leads to local alternatives of order $n^{-1/4}$. The
scalar case result is already familiar from existing literature and we extend
it to the multi-dimensional case. The new framework reveals that certain
regularity conditions commonly employed in earlier developments are
unnecessary, i.e. the symmetry or third moment condition imposed on the
heterogeneity distribution. Additionally, the limit experiment for the
multi-dimensional case suggests modifications on existing tests for slope
heterogeneity in cross sectional and panel data models that lead to power
improvement. Since the $C(\alpha)$ framework is not restricted to the
parametric model and the test statistics do not depend on the particular choice
of the heterogeneity distribution, it is useful for a broad range of
applications for testing parametric heterogeneity.
