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Asymptotics for Markov chain mixture detection

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  • Fitzpatrick, Matthew
  • Stewart, Michael

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-state-space 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.

Suggested Citation

  • Fitzpatrick, Matthew & Stewart, Michael, 2022. "Asymptotics for Markov chain mixture detection," Econometrics and Statistics, Elsevier, vol. 22(C), pages 56-66.
    • Handle: RePEc:eee:ecosta:v:22:y:2022:i:c:p:56-66
    DOI: 10.1016/j.ecosta.2021.11.004
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    References listed on IDEAS

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    1. Xin Liu & Cristian Pasarica & Yongzhao Shao, 2003. "Testing Homogeneity in Gamma Mixture Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(1), pages 227-239, March.
    2. Frydman, Halina, 2005. "Estimation in the Mixture of Markov Chains Moving With Different Speeds," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 1046-1053, September.
    3. Yuhong Yang, 2005. "Can the strengths of AIC and BIC be shared? A conflict between model indentification and regression estimation," Biometrika, Biometrika Trust, vol. 92(4), pages 937-950, December.
    4. Francis K.C. Hui & David I. Warton & Scott D. Foster, 2015. "Order selection in finite mixture models: complete or observed likelihood information criteria?," Biometrika, Biometrika Trust, vol. 102(3), pages 724-730.
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