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Comparison of Nonparametric Estimators for the Renewal Function

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  • Helmut Schneider
  • Bin‐Shan Lin
  • Colm O'Cinneide

Abstract

This paper addresses the problem of nonparametric estimation of the renewal function. Two estimators are discussed. The first estimator, introduced by Frees, is based on the sum of the ‘convolutions without replacement’ of the empirical distribution function. We suggest a polynomial time algorithm to compute this estimator. The second estimator is based on the renewal function of the empirical distribution. We show how this estimator may be computed efficiently by solving a discretized renewal equation. In a simulation study we show that the estimator based on the renewal equation has a slightly higher bias than the estimator introduced by Frees, while the mean‐squared errors are about the same. However, the computing time for the new estimator suggested in this paper is generally much smaller than that for Frees's estimator.

Suggested Citation

  • Helmut Schneider & Bin‐Shan Lin & Colm O'Cinneide, 1990. "Comparison of Nonparametric Estimators for the Renewal Function," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 39(1), pages 55-61, March.
    • Handle: RePEc:bla:jorssc:v:39:y:1990:i:1:p:55-61
    DOI: 10.2307/2347811
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    Cited by:

    1. Marlin U. Thomas & Sridevi S. Rao, 1999. "Warranty Economic Decision Models: A Summary and Some Suggested Directions for Future Research," Operations Research, INFORMS, vol. 47(6), pages 807-820, December.
    2. Altındağ, Ömer & Aydoğdu, Halil, 2021. "Estimation of renewal function under progressively censored data and its applications," Reliability Engineering and System Safety, Elsevier, vol. 216(C).
    3. Emmanuel Hagenimana & Song Lixin & Patrick Kandege, 2018. "Study of nonparametric estimation details of instant system availability average," International Journal of System Assurance Engineering and Management, Springer;The Society for Reliability, Engineering Quality and Operations Management (SREQOM),India, and Division of Operation and Maintenance, Lulea University of Technology, Sweden, vol. 9(2), pages 467-481, April.
    4. Brahim Ouhbi & Nikolaos Limnios, 1999. "Nonparametric Estimation for Semi-Markov Processes Based on its Hazard Rate Functions," Statistical Inference for Stochastic Processes, Springer, vol. 2(2), pages 151-173, May.
    5. Steven G. From & Linxiong Li, 2003. "Nonparametric estimation of some quantities of interest from renewal theory," Naval Research Logistics (NRL), John Wiley & Sons, vol. 50(6), pages 638-649, September.

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