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ML, PL, QL in Markov Chain Models

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  • NILS LID HJORT
  • CRISTIANO VARIN

Abstract

. In many spatial and spatial‐temporal models, and more generally in models with complex dependencies, it may be too difficult to carry out full maximum‐likelihood (ML) analysis. Remedies include the use of pseudo‐likelihood (PL) and quasi‐likelihood (QL) (also called the composite likelihood). The present paper studies the ML, PL and QL methods for general Markov chain models, partly motivated by the desire to understand the precise behaviour of the PL and QL methods in settings where this can be analysed. We present limiting normality results and compare performances in different settings. For Markov chain models, the PL and QL methods can be seen as maximum penalized likelihood methods. We find that QL is typically preferable to PL, and that it loses very little to ML, while sometimes earning in model robustness. It has also appeal and potential as a modelling tool. Our methods are illustrated for consonant‐vowel transitions in poetry and for analysis of DNA sequence evolution‐type models.

Suggested Citation

  • 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.
    • Handle: RePEc:bla:scjsta:v:35:y:2008:i:1:p:64-82
    DOI: 10.1111/j.1467-9469.2007.00559.x
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    1. Hobolth Asger & Jensen Jens Ledet, 2005. "Statistical Inference in Evolutionary Models of DNA Sequences via the EM Algorithm," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 4(1), pages 1-22, August.
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    Cited by:

    1. Anders Løland & Ragnar Bang Huseby & Nils Lid Hjort & Arnoldo Frigessi, 2013. "Statistical Corrections of Invalid Correlation Matrices," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(4), pages 807-824, December.
    2. Ioulia Papageorgiou, 2016. "Sampling from Correlated Populations: Optimal Strategies and Comparison Study," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 78(1), pages 119-151, May.
    3. Kenne Pagui, E.C. & Salvan, A. & Sartori, N., 2015. "On full efficiency of the maximum composite likelihood estimator," Statistics & Probability Letters, Elsevier, vol. 97(C), pages 120-124.
    4. Cristiano Varin, 2008. "On composite marginal likelihoods," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 92(1), pages 1-28, February.
    5. Bhat, Chandra R., 2011. "The maximum approximate composite marginal likelihood (MACML) estimation of multinomial probit-based unordered response choice models," Transportation Research Part B: Methodological, Elsevier, vol. 45(7), pages 923-939, August.
    6. Emil Aas Stoltenberg & Nils Lid Hjort, 2021. "Models and inference for on–off data via clipped Ornstein–Uhlenbeck processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(3), pages 908-929, September.
    7. Bhat, Chandra R. & Sener, Ipek N. & Eluru, Naveen, 2010. "A flexible spatially dependent discrete choice model: Formulation and application to teenagers' weekday recreational activity participation," Transportation Research Part B: Methodological, Elsevier, vol. 44(8-9), pages 903-921, September.
    8. 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.
    9. Paleti, Rajesh & Bhat, Chandra R., 2013. "The composite marginal likelihood (CML) estimation of panel ordered-response models," Journal of choice modelling, Elsevier, vol. 7(C), pages 24-43.
    10. Bartolucci, Francesco & Lupparelli, Monia, 2012. "Nested hidden Markov chains for modeling dynamic unobserved heterogeneity in multilevel longitudinal data," MPRA Paper 40588, University Library of Munich, Germany.

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