  This research creates a general class of "perturbation models" which are
described by an underlying "null" model that accounts for most of the structure
in data and a perturbation that accounts for possible small localized
departures. The perturbation models encompass finite mixture models and spatial
scan process. In this article, (1) we propose a new test statistic to detect
the presence of perturbation, including the case where the null model contains
a set of nuisance parameters, and show that it is equivalent to the likelihood
ratio test; (2) we establish that the asymptotic distribution of the test
statistic is equivalent to the supremum of a Gaussian random field over a
high-dimensional manifold (e.g., curve, surface etc.) with boundaries and
singularities; (3) we derive a technique for approximating the quantiles of the
test statistic using the Hotelling-Weyl-Naiman "volume-of-tube formula"; and
(4) we solve the long-pending problem of testing for the order of a mixture
model; in particular, derive the asymptotic null distribution for a general
family of mixture models including the multivariate mixtures. The inferential
theory developed in this article is applicable for a class of non-regular
statistical problems involving loss of identifiability or when some of the
parameters are on the boundary of the parametric space.
