The Volume-of-Tube Formula: Perturbation Tests, Mixture Models and Scan Statistics

This article creates a general class of perturbation models which are described by an underlying null model that accounts for most of the structure in data while a perturbation accounts for possible small localized departures. The goal is to develop theory and inferential methods for fitting the perturbation models including the general case when the null model contains a set of nuisance parameters. Applications of the perturbation models include finite mixture models and spatial scan process. In the context of finite mixture models, the null model represents a mixture with m components and the perturbation model represents additional components. In the spatial scan process context, the null density accounts for the background or noise whereas the perturbation searches for an unusual region such as a tumorous tissue in mammography or a target in an automatic target recognition problem. In this research, (1) a test statistic for detecting the presence of perturbation is proposed and is shown to be equivalent to the likelihood ratio test in certain cases, (2) a general theory for the derivation of the limiting distribution of the test statistic for a class of non-regular problems is developed using the Hotelling-Weyl volume-of-tube formula, (3) the resulting theory is applied to the problem of testing for the order of a mixture model; in particular, the limiting distribution is derived for a general family of mixture models including the multivariate mixture case and (4) explicit determination of the geometric constants appearing in the tube formula is developed.