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Robust parametric inference for finite Markov chains

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  • Abhik Ghosh

    (Indian Statistical Institute)

Abstract

We consider the problem of statistical inference in a parametric finite Markov chain model and develop a robust estimator of the parameters defining the transition probabilities via minimization of a suitable (empirical) version of the popular density power divergence. Based on a long sequence of observations from a first-order stationary Markov chain, we have defined the minimum density power divergence estimator (MDPDE) of the underlying parameter and rigorously derived its asymptotic and robustness properties under appropriate conditions. Performance of the MDPDEs is illustrated theoretically as well as empirically for some common examples of finite Markov chain models. Its applications in robust testing of statistical hypotheses are also discussed along with (parametric) comparison of two Markov chain sequences. Several directions for extending the MDPDE and related inference are also briefly discussed for multiple sequences of Markov chains, higher order Markov chains and non-stationary Markov chains with time-dependent transition probabilities. Finally, our proposal is applied to analyze corporate credit rating migration data of three international markets.

Suggested Citation

  • Abhik Ghosh, 2022. "Robust parametric inference for finite Markov chains," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(1), pages 118-147, March.
    • Handle: RePEc:spr:testjl:v:31:y:2022:i:1:d:10.1007_s11749-021-00771-1
    DOI: 10.1007/s11749-021-00771-1
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    References listed on IDEAS

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    1. Ghosh, Abhik & Mandal, Abhijit & Martín, Nirian & Pardo, Leandro, 2016. "Influence analysis of robust Wald-type tests," Journal of Multivariate Analysis, Elsevier, vol. 147(C), pages 102-126.
    2. Linda Möstel & Marius Pfeuffer & Matthias Fischer, 2020. "Statistical inference for Markov chains with applications to credit risk," Computational Statistics, Springer, vol. 35(4), pages 1659-1684, December.
    3. Nils Lid Hjort & Cristiano Varin, 2008. "ML, PL, QL in Markov Chain Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 35(1), pages 64-82, March.
    4. Ayanendranath Basu & Abhik Ghosh & Nirian Martin & Leandro Pardo, 2018. "Robust Wald-type tests for non-homogeneous observations based on the minimum density power divergence estimator," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 81(5), pages 493-522, July.
    5. Srabashi Basu & Ayanendranath Basu & M. Jones, 2006. "Robust and Efficient Parametric Estimation for Censored Survival Data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 58(2), pages 341-355, June.
    6. Sangyeol Lee & Junmo Song, 2013. "Minimum density power divergence estimator for diffusion processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 65(2), pages 213-236, April.
    7. L. Zhao & C. Dorea & C. Gonçalves, 2001. "On Determination of the Order of a Markov Chain," Statistical Inference for Stochastic Processes, Springer, vol. 4(3), pages 273-282, October.
    8. Menéndez, M. L. & Morales, D. & Pardo, L. & Zografos, K., 1999. "Statistical inference for finite Markov chains based on divergences," Statistics & Probability Letters, Elsevier, vol. 41(1), pages 9-17, January.
    9. Kang, Jiwon & Lee, Sangyeol, 2014. "Minimum density power divergence estimator for Poisson autoregressive models," Computational Statistics & Data Analysis, Elsevier, vol. 80(C), pages 44-56.
    10. Papapetrou, M. & Kugiumtzis, D., 2013. "Markov chain order estimation with conditional mutual information," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(7), pages 1593-1601.
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