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Robust parametric inference for finite Markov chains

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.

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References

  • Anderson TW, Goodman LA (1957) Statistical inference about Markov chains. Ann Math Stat 89–110

  • Basu, S., Basu, A. and Jones, M. C. (2006). Robust and efficient parametric estimation for censored survival data. Ann Inst Stat Math, 58(2), 341–355

    MathSciNet  Article  Google Scholar 

  • Basu, A., Ghosh, A., Martin, N. and Pardo, L. (2018) Robust Wald-type tests for non-homogeneous observations based on minimum density power divergence estimator. Metrika, 81, 493–522

    MathSciNet  Article  Google Scholar 

  • Basu A, Ghosh A, Martin N, Pardo L (2021) A robust generalization of the Rao test. J Bus Econ Stat. https://doi.org/10.1080/07350015.2021.1876711

  • Basu, A., Harris, I. R., Hjort, N. L., and Jones, M. C. (1998). Robust and efficient estimation by minimising a density power divergence. Biometrika 85, 549–559

    MathSciNet  Article  Google Scholar 

  • Basu A, Shioya H, Park C (2011) Statistical Inference: The Minimum Distance Approach. Chapman & Hall/CRC, Boca Raton, Florida

    Book  Google Scholar 

  • Bertail P, Ciolek G, Tillier C (2018) Robust estimation for Markov chains with applications to piecewise-deterministic Markov processes. Statistical inference for piecewise-deterministic Markov processes. Wiley Online Library, pp 107–146

  • Billingsley P (1961) Statistical methods in Markov chains. Ann Math Stat 12–40

  • Birge L (1983) Robust testing for independent non identically distributed variables and Markov chains Specifying statistical models. Springer, Berlin, pp 134–162

    Google Scholar 

  • Gani, J., and Jerwood, D. (1971). Markov chain methods in chain binomial epidemic models. Biometrics, 27, 591–603

    Article  Google Scholar 

  • Ghosh, A. and Basu, A. (2013). Robust estimation for independent non-homogeneous observations using density power divergence with applications to linear regression. Electron. J Stat, 7, 2420–2456

    MathSciNet  Article  Google Scholar 

  • Ghosh A, Basu A (2018) Robust bounded influence tests for independent but non-homogeneous observations. Sta. Sin 28(3):1133–1155

    MATH  Google Scholar 

  • Ghosh, A., Mandal, A., Martin, N. and Pardo, L. (2016). Influence analysis of robust Wald-type tests. J Mult Anal, 147, 102–126

    MathSciNet  Article  Google Scholar 

  • Ghosh A, Martin N, Basu A, Pardo L (2018) A new class of robust two-sample Wald-type tests. Int J Biostat 14(2)

  • Hampel FR, Ronchetti E, Rousseeuw PJ, Stahel W (1986) Robust statistics: the approach based on influence functions. Wiley, New York

    MATH  Google Scholar 

  • Hjort, N.L., and Varin, C. (2008). ML, PL, QL in Markov chain models. Scand J Stat, 35, 64–82

    MathSciNet  Article  Google Scholar 

  • Iosifescu M (2007) Finite Markov Processes and Their Applications. Dover Publications Inc., NY, USA

    MATH  Google Scholar 

  • Jones GL (2004) On the Markov chain central limit theorem. Prob Surveys 1(299–320):5–1

    MathSciNet  Article  Google Scholar 

  • Kang, J., and Lee, S. (2014). Minimum density power divergence estimator for Poisson autoregressive models. Comput Stat Data Anal, 80, 44–56

    MathSciNet  Article  Google Scholar 

  • Kunsch H (1984) Infinitesimal Robustness for Autoregressive Processes. Ann Stat 12(3):843–863

    MathSciNet  Article  Google Scholar 

  • Lee, S. and Lee, T. (2010). Robust estimation for order of hidden Markov models based on density power divergences. J Stat Comput Simul, 80(5), 503–512

    MathSciNet  Article  Google Scholar 

  • Lee, S., and Song, J. (2013). Minimum density power divergence estimator for diffusion processes. Ann Inst Stat Math, 65(2), 213–236

    MathSciNet  Article  Google Scholar 

  • Lifshits, B. A. (1979). On the central limit theorem for Markov chains. Theory Prob Appl, 23(2), 279–296

    Article  Google Scholar 

  • Martin RD, Yohai VJ (1986) Influence Functionals for Time Series. Ann Stat 1986(14):781–818

    MathSciNet  MATH  Google Scholar 

  • Menendez ML, Morales D, Pardo L, Zografos K (1999) Statistical inference for finite Markov chains based on divergences. Stat Prob Letters 41(1):9–17

    MathSciNet  Article  Google Scholar 

  • Menendez ML, Pardo JA, Pardo L (2001) Csiszar’s \(\phi \)-divergences for testing the order in a Markov chain. Stat Pap 42(3):313–328

  • Mostel, L., Pfeuffer, M., and Fischer, M. (2020). Statistical inference for Markov chains with applications to credit risk. Comput Stat, 35, 1659–1684

    MathSciNet  Article  Google Scholar 

  • Papapetrou, M., and Kugiumtzis, D. (2013). Markov chain order estimation with conditional mutual information. Physica A, 392(7), 1593–1601

    MathSciNet  Article  Google Scholar 

  • Rajarshi MB (2014) Statistical inference for discrete time stochastic processes. Springer

    MATH  Google Scholar 

  • Richhariya NM, Jain M, Debnath A et al (2019) Default, transition, and recovery: 2018 annual global corporate default and rating transition study. Standard & poor’s ratings direct. https://www.spratings.com/documents/20184/774196/2018AnnualGlobalCorporateDefaultAndRatingTransitionStudy.pdf

  • Sirazhdinov, S. K., and Formanov, S. K. (1984). On estimates of the rate of convergence in the central limit theorem for homogeneous Markov chains. Theory Prob Appl, 28(2), 229–239

    MathSciNet  Article  Google Scholar 

  • Zhao, L. C., Dorea, C. C. Y., and Gonçalves, C. R. (2001). On determination of the order of a Markov chain. Stat Inf Stoch Proc, 4(3), 273–282

    MathSciNet  Article  Google Scholar 

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Acknowledgements

The author wishes to thank the editor, the associate editor and two anonymous referees for their careful reading of the manuscript and several constructive suggestions to improve the paper. This research is partially supported by the INSPIRE Faculty Research Grant from Department of Science and Technology, Government of India.

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Correspondence to Abhik Ghosh.

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Ghosh, A. Robust parametric inference for finite Markov chains. TEST 31, 118–147 (2022). https://doi.org/10.1007/s11749-021-00771-1

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  • DOI: https://doi.org/10.1007/s11749-021-00771-1

Keywords

  • Minimum density power divergence estimator
  • Finite Markov chain
  • Parametric inference
  • Robustness

Mathematics Subject Classification

  • Primary 62F35
  • 62M02
  • Secondary 60J20