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A Phase-Type Distribution for the Sum of Two Concatenated Markov Processes Application to the Analysis Survival in Bladder Cancer

Author

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  • Belén García-Mora

    (Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, 46022 Valencia, Spain)

  • Cristina Santamaría

    (Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, 46022 Valencia, Spain)

  • Gregorio Rubio

    (Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, 46022 Valencia, Spain)

Abstract

Stochastic processes are useful and important for modeling the evolution of processes that take different states over time, a situation frequently found in fields such as medical research and engineering. In a previous paper and within this framework, we developed the sum of two independent phase-type (PH)-distributed variables, each of them being associated with a Markovian process of one absorbing state. In that analysis, we computed the distribution function, and its associated survival function, of the sum of both variables, also PH-distributed. In this work, in one more step, we have developed a first approximation of that distribution function in order to avoid the calculation of an inverse matrix for the possibility of a bad conditioning of the matrix, involved in the expression of the distribution function in the previous paper. Next, in a second step, we improve this result, giving a second, more accurate approximation. Two numerical applications, one with simulated data and the other one with bladder cancer data, are used to illustrate the two proposed approaches to the distribution function. We compare and argue the accuracy and precision of each one of them by means of their error bound and the application to real data of bladder cancer.

Suggested Citation

  • Belén García-Mora & Cristina Santamaría & Gregorio Rubio, 2020. "A Phase-Type Distribution for the Sum of Two Concatenated Markov Processes Application to the Analysis Survival in Bladder Cancer," Mathematics, MDPI, vol. 8(12), pages 1-15, November.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:12:p:2099-:d:450097
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    References listed on IDEAS

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    1. Jackson, Christopher, 2011. "Multi-State Models for Panel Data: The msm Package for R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 38(i08).
    2. Mark Fackrell, 2012. "A semi-infinite programming approach to identifying matrix-exponential distributions," International Journal of Systems Science, Taylor & Francis Journals, vol. 43(9), pages 1623-1631.
    3. Mark Fackrell, 2009. "Modelling healthcare systems with phase-type distributions," Health Care Management Science, Springer, vol. 12(1), pages 11-26, March.
    4. Montoro-Cazorla, Delia & Pérez-Ocón, Rafael, 2014. "Matrix stochastic analysis of the maintainability of a machine under shocks," Reliability Engineering and System Safety, Elsevier, vol. 121(C), pages 11-17.
    5. Rodríguez, Joanna & Lillo, Rosa E. & Ramírez-Cobo, Pepa, 2015. "Failure modeling of an electrical N-component framework by the non-stationary Markovian arrival process," Reliability Engineering and System Safety, Elsevier, vol. 134(C), pages 126-133.
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