Abstract
Methods of estimating unknown parameters of a trend function for trend-renewal processes are investigated in the case when the renewal distribution function is unknown. If the renewal distribution is unknown, then the likelihood function of the trend-renewal process is unknown and consequently the maximum likelihood method cannot be used. In such a situation we propose three other methods of estimating the trend parameters. The methods proposed can also be used to predict future occurrence times. The performance of the estimators based on these methods is illustrated numerically for some trend-renewal processes for which the statistical inference is analytically intractable.
Article PDF
Similar content being viewed by others
References
Andersen, P.K., Borgan, Ø., Gill, R.D., Keiding, N.: Statistical Models Based on Counting Processes. Springer Series in Statistics. Springer, New York (1993)
Heggland, K., Lindqvist, B.: A non-parametric monotone maximum likelihood estimator of time trend for repairable system data. Reliab. Eng. Syst. Saf. 92, 575–584 (2007)
Khoshgoftaar, T.M.: Nonhomogeneous Poisson processes for software reliability growth. In: COMPSTAT’88, Copenhagen, Denmark (1988)
Lindqvist, B.: The trend-renewal process, a useful model for repairable systems. Malmö, Sweden, Society in Reliability Engineers, Scandinavian Chapter, Annual Conference (1993)
Lindqvist, B., Elvebakk, G., Heggland, K.: The trend-renewal process for statistical analysis of repairable systems. Technometrics 45(1), 31–44 (2003)
Lindqvist, B., Kjønstad, G., Meland, N.: Testing for trend in repairable system data. In: Proceedings of ESREL’94, La Boule, France (1994)
Lindqvist, B.H.: On the statistical modeling and analysis of repairable systems. Stat. Sci. 21(4), 532–551 (2006)
Lindqvist, B.H., Doksum, K.A. (eds.): Mathematical and Statistical Methods in Reliability. Series on Quality, Reliability & Engineering Statistics, vol. 7. World Scientific, River Edge (2003). Papers from the 3rd International Conference in Mathematical Methods in Reliability held in Trondheim, June 17–20, 2002
Nayak, T., Bose, S., Kundu, S.: On inconsistency of estimators of parameters of non-homogeneous Poisson process models for software reliability. Stat. Probab. Lett. 78, 2217–2221 (2008)
Peña, E., Hollander, M.: Models for recurrent events in reliability and survival analysis. In: Soyer, R., Mazzuchi, T., Singpurwalla, N. (eds.) Mathematical Reliability: An Expository Perspective, pp. 105–118. Kluwer Academic, Dordrecht (2004)
Rigdon, S., Basu, A.: Statistical Methods for the Reliability of Repairable Systems. Wiley, New York (2000)
Stocker, R., Peña, E.: A general class of parametric models for recurrent event data. Technometrics 49(2), 210–220 (2007)
Zhao, M., Xie, M.: On maximum likelihood estimation for a general non-homogeneous Poisson process. Scand. J. Stat. 23(4), 597–607 (1996)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Jokiel-Rokita, A., Magiera, R. Estimation of parameters for trend-renewal processes. Stat Comput 22, 625–637 (2012). https://doi.org/10.1007/s11222-011-9260-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11222-011-9260-1