This research creates a general class of "perturbation models" which aredescribed by an underlying "null" model that accounts for most of the structurein data and a perturbation that accounts for possible small localizeddepartures. The perturbation models encompass finite mixture models and spatialscan process. In this article, (1) we propose a new test statistic to detectthe presence of perturbation, including the case where the null model containsa set of nuisance parameters, and show that it is equivalent to the likelihoodratio test; (2) we establish that the asymptotic distribution of the teststatistic is equivalent to the supremum of a Gaussian random field over ahigh-dimensional manifold (e.g., curve, surface etc.) with boundaries andsingularities; (3) we derive a technique for approximating the quantiles of thetest 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 mixturemodel; in particular, derive the asymptotic null distribution for a generalfamily of mixture models including the multivariate mixtures. The inferentialtheory developed in this article is applicable for a class of non-regularstatistical problems involving loss of identifiability or when some of theparameters are on the boundary of the parametric space.
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