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Kernel estimators of Markov renewal and semi-Markov transition functions of semi-Markov systems

Author

Listed:
  • Fatiha Mokhtari
  • Chafiâa Ayhar
  • Vlad Stefan Barbu
  • Saâdia Rahmani

Abstract

The Markov renewal and semi-Markov transition functions are of considerable importance when studying the behaviour of semi-Markov processes, or equivalently Markov renewal processes. In this paper, we introduce nonparametric kernel estimators of these two functions, for continuous-time, homogeneous semi-Markov process. The main results given in this paper are the asymptotic properties of these estimators, as the strong consistency and the asymptotic normality. A numerical example is provided to support the theoretical results. At this stage, we also address the problem of numerically solving the continuous-time Markov renewal equation, together with an associated algorithm. This is a crucial and non-trivial point, in order to practically find the solution of continuous-time Markov renewal equation since the numerical complexity of such equations becomes very fast intractable in practice.

Suggested Citation

  • Fatiha Mokhtari & Chafiâa Ayhar & Vlad Stefan Barbu & Saâdia Rahmani, 2026. "Kernel estimators of Markov renewal and semi-Markov transition functions of semi-Markov systems," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 38(2), pages 421-444, April.
    • Handle: RePEc:taf:gnstxx:v:38:y:2026:i:2:p:421-444
    DOI: 10.1080/10485252.2025.2490939
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