Order selection with confidence for finite mixture models

TL;DR

This paper introduces a sequential testing procedure based on the closed testing principle for confidently determining the number of components in finite mixture models, supported by finite sample tests and asymptotic consistency.

Contribution

It develops a novel sequential testing framework with finite sample tests for mixture model order selection and confidence statements, extending applicability to nested models.

Findings

Finite sample tests are consistent against fixed alternatives.

The proposed STP provides reliable confidence statements about model order.

A modified STP achieves asymptotic consistency in order selection.

Abstract

The determination of the number of mixture components (the order) of a finite mixture model has been an enduring problem in statistical inference. We prove that the closed testing principle leads to a sequential testing procedure (STP) that allows for confidence statements to be made regarding the order of a finite mixture model. We construct finite sample tests, via data splitting and data swapping, for use in the STP, and we prove that such tests are consistent against fixed alternatives. Simulation studies and real data examples are used to demonstrate the performance of the finite sample tests-based STP, yielding practical recommendations of their use as confidence estimators in combination with point estimates such as the Akaike information or Bayesian information criteria. In addition, we demonstrate that a modification of the STP yields a method that consistently selects the order of a finite mixture model, in the asymptotic sense. Our STP is not only applicable for order selection of finite mixture models, but is also useful for making confidence statements regarding any sequence of nested models.