Inference in Perturbation Models, Finite Mixtures and Scan Statistics: The Volume-of-Tube Formula

TL;DR

This paper develops a unified approach for inference in perturbation models, including finite mixtures and scan statistics, using the volume-of-tube formula to approximate distributions of test statistics.

Contribution

It introduces a new likelihood ratio-based test for perturbation detection, derives its asymptotic distribution, and applies the volume-of-tube formula for quantile approximation, solving longstanding mixture model testing issues.

Findings

The test statistic's distribution is linked to Gaussian random fields over manifolds.

The volume-of-tube formula effectively approximates critical values for complex models.

The method addresses non-regular problems with boundary or identifiability issues.

Abstract

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.