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Estimating the inter-arrival time density of Markov renewal processes under structural assumptions on the transition distribution


  • Greenwood, Priscilla E.
  • Schick, Anton
  • Wefelmeyer, Wolfgang


We consider a stationary Markov renewal process whose inter-arrival time density depends multiplicatively on the distance between the past and present state of the embedded chain. This is appropriate when the jump size is governed by influences that accumulate over time. Then we can construct an estimator for the inter-arrival time density that has the parametric rate of convergence. The estimator is a local von Mises statistic. The result carries over to the corresponding semi-Markov process.

Suggested Citation

  • Greenwood, Priscilla E. & Schick, Anton & Wefelmeyer, Wolfgang, 2011. "Estimating the inter-arrival time density of Markov renewal processes under structural assumptions on the transition distribution," Statistics & Probability Letters, Elsevier, vol. 81(2), pages 277-282, February.
  • Handle: RePEc:eee:stapro:v:81:y:2011:i:2:p:277-282

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    References listed on IDEAS

    1. Meerschaert, Mark M. & Scalas, Enrico, 2006. "Coupled continuous time random walks in finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 370(1), pages 114-118.
    2. Zudi Lu, 2001. "Asymptotic Normality of Kernel Density Estimators under Dependence," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(3), pages 447-468, September.
    3. Schick Anton & Wefelmeyer Wolfgang, 2009. "Non-standard behavior of density estimators for sums of squared observations," Statistics & Risk Modeling, De Gruyter, vol. 27(1), pages 55-73, November.
    4. Anton Schick & Wolfgang Wefelmeyer, 2008. "Root-n consistency in weighted L 1 -spaces for density estimators of invertible linear processes," Statistical Inference for Stochastic Processes, Springer, vol. 11(3), pages 281-310, October.
    5. Glen, Andrew G. & Leemis, Lawrence M. & Drew, John H., 2004. "Computing the distribution of the product of two continuous random variables," Computational Statistics & Data Analysis, Elsevier, vol. 44(3), pages 451-464, January.
    6. Raberto, Marco & Scalas, Enrico & Mainardi, Francesco, 2002. "Waiting-times and returns in high-frequency financial data: an empirical study," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 314(1), pages 749-755.
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