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Two-sided Bounds for some Quantities in the Delayed Renewal Process

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  • Stathis Chadjiconstantinidis

    (University of Piraeus)

Abstract

In this paper we obtain some “general” two-sided bounds for the delayed renewal function, in the sense that the bounds are valid for any arbitrary distributions of the inter-arrival times. Also, we give a sequence of monotone non-decreasing (non-increasing) lower (upper) general bounds converging to the delayed renewal function. By considering several aging or reliability classes for the distribution of the interarrival times (e.g., $$DFR$$ DFR , bounded mean residual lifetime, $$NBUE$$ NBUE , $$NWUE$$ NWUE , bounded failure rate, $$DMRL$$ DMRL , $$IMRL$$ IMRL ) we give upper and lower bounds for the delayed renewal function, and moreover by assuming the usual stochastic order between the first and the subsequent interarrival times, we give sequences of monotone non-decreasing (non-increasing) lower (upper) bounds converging to the delayed renewal function. Also, some sequences of bounds for the delayed renewal function in terms of the ordinary renewal function are given. Sequences of monotone non-decreasing (non-increasing) lower (upper) bounds for the delayed renewal density are also given. Finally, we obtain upper and lower bounds for the expected number of renewals over a finite interval, and as a result, we get an improvement of the upper bounds obtained by Lorden (Ann Math Statist 41:520–527, 1970) and Losidis and Politis (2022) for the expected number of renewals over a finite interval under the ordinary renewal process.

Suggested Citation

  • Stathis Chadjiconstantinidis, 2024. "Two-sided Bounds for some Quantities in the Delayed Renewal Process," Methodology and Computing in Applied Probability, Springer, vol. 26(3), pages 1-48, September.
    • Handle: RePEc:spr:metcap:v:26:y:2024:i:3:d:10.1007_s11009-024-10088-9
    DOI: 10.1007/s11009-024-10088-9
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    References listed on IDEAS

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    1. Cuffe, Barry P. & Friedman, Moshe F., 2006. "On the exact distribution of a delayed renewal process with exponential sum interarrival times," European Journal of Operational Research, Elsevier, vol. 172(3), pages 909-918, August.
    2. Richard M. Soland, 1968. "A Renewal Theoretic Approach to the Estimation of Future Demand for Replacement Parts," Operations Research, INFORMS, vol. 16(1), pages 36-51, February.
    3. Sotirios Losidis & Konstadinos Politis, 2022. "Bounds for the Renewal Function and Related Quantities," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2647-2660, December.
    4. Losidis, Sotirios & Politis, Konstadinos, 2017. "A two-sided bound for the renewal function when the interarrival distribution is IMRL," Statistics & Probability Letters, Elsevier, vol. 125(C), pages 164-170.
    5. Mustafa Hilmi Pekalp & Ömer Altındağ & Özgür Acar & Halil Aydoğdu, 2020. "Plug-in estimators for the mean value and variance functions in delayed renewal processes," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 49(19), pages 4693-4711, October.
    6. Cai, Jun & Garrido, Jose, 1998. "Aging properties and bounds for ruin probabilities and stop-loss premiums," Insurance: Mathematics and Economics, Elsevier, vol. 23(1), pages 33-43, October.
    7. Dermitzakis, Vaios & Politis, Konstadinos, 2022. "Monotonicity properties for solutions of renewal equations," Statistics & Probability Letters, Elsevier, vol. 180(C).
    Full references (including those not matched with items on IDEAS)

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