Projected likelihood contrasts for testing homogeneity in finite mixture models with nuisance parameters

TL;DR

This paper introduces a new statistical test called Projected Likelihood Contrast for assessing homogeneity in finite mixture models with nuisance parameters, supported by theoretical analysis and simulation studies.

Contribution

It develops the PLC test, a modification of likelihood ratio and Rao's score tests, specifically designed for mixture models with known mixing proportions and nuisance parameters.

Findings

PLC tests have well-understood large sample properties.

Simulation results confirm the theoretical behavior of the PLC statistic.

Power analysis shows effectiveness of PLC in Gaussian mixture scenarios.

Abstract

This paper develops a test for homogeneity in finite mixture models where the mixing proportions are known a priori (taken to be 0.5) and a common nuisance parameter is present. Statistical tests based on the notion of Projected Likelihood Contrasts (PLC) are considered. The PLC is a slight modification of the usual likelihood ratio statistic or the Wilk's $\Lambda$ and is similar in spirit to the Rao's score test. Theoretical investigations have been carried out to understand the large sample statistical properties of these tests. Simulation studies have been carried out to understand the behavior of the null distribution of the PLC statistic in the case of Gaussian mixtures with unknown means (common variance as nuisance parameter) and unknown variances (common mean as nuisance parameter). The results are in conformity with the theoretical results obtained. Power functions of these tests have been evaluated based on simulations from Gaussian mixtures.