Asymptotics for Markov chain mixture detection

TL;DR

This paper investigates the asymptotic behavior of the likelihood ratio test in Markov chain mixture models, revealing conditions where the test statistic diverges and deriving its limiting distribution in specific cases.

Contribution

It provides new theoretical conditions under which the likelihood ratio test does not follow the chi-squared distribution, especially for Markov chain mixture models, and derives the limiting distribution in a special case.

Findings

Likelihood ratio statistic diverges under certain conditions.

Derived Gumbel distribution for a specific two-state Markov mixture.

Validated results with bootstrap simulation on bond ratings.

Abstract

Sufficient conditions are provided under which the log-likelihood ratio test statistic fails to have a limiting chi-squared distribution under the null hypothesis when testing between one and two components under a general two-component mixture model, but rather tends to infinity in probability. These conditions are verified when the component densities describe continuous-time, discrete-statespace Markov chains and the results are illustrated via a parametric bootstrap simulation on an analysis of the migrations over time of a set of corporate bonds ratings. The precise limiting distribution is derived in a simple case with two states, one of which is absorbing which leads to a right-censored exponential scale mixture model. In that case, when centred by a function growing logarithmically in the sample size, the statistic has a limiting distribution of Gumbel extreme-value type rather than chi-squared.