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Nonparametric estimation in trend-renewal processes

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  • Gámiz, Maria Luz
  • Lindqvist, Bo Henry

Abstract

The trend-renewal-process (TRP) is defined to be a time-transformed renewal process, where the time transformation is given by a trend function λ(·) which is similar to the intensity of a nonhomogeneous Poisson process (NHPP). A nonparametric maximum likelihood estimator of the trend function of a TRP can be obtained in principle in a similar manner as for the NHPP using kernel smoothing. For a full nonparametric estimation of a trend-renewal process it is necessary, however, to estimate jointly the trend function and the renewal distribution. For this purpose we consider a nonparametric approach using kernel smoothing techniques. We develop an original algorithm to estimate the conditional intensity function by preserving its structure in terms of the trend function and the underlying renewal process. The algorithm is applied to both simulated and real data sets.

Suggested Citation

  • Gámiz, Maria Luz & Lindqvist, Bo Henry, 2016. "Nonparametric estimation in trend-renewal processes," Reliability Engineering and System Safety, Elsevier, vol. 145(C), pages 38-46.
    • Handle: RePEc:eee:reensy:v:145:y:2016:i:c:p:38-46
    DOI: 10.1016/j.ress.2015.08.015
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    References listed on IDEAS

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    1. Feng Chen & Paul S. F. Yip & K. F. Lam, 2011. "On the Local Polynomial Estimators of the Counting Process Intensity Function and its Derivatives," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 38(4), pages 631-649, December.
    2. Tanwar, Monika & Rai, Rajiv N. & Bolia, Nomesh, 2014. "Imperfect repair modeling using Kijima type generalized renewal process," Reliability Engineering and System Safety, Elsevier, vol. 124(C), pages 24-31.
    3. Heggland, Knut & Lindqvist, Bo H., 2007. "A non-parametric monotone maximum likelihood estimator of time trend for repairable system data," Reliability Engineering and System Safety, Elsevier, vol. 92(5), pages 575-584.
    4. Luo, Xiaopeng & Lu, Zhenzhou & Xu, Xin, 2014. "Non-parametric kernel estimation for the ANOVA decomposition and sensitivity analysis," Reliability Engineering and System Safety, Elsevier, vol. 130(C), pages 140-148.
    5. M. Luz Gámiz & K. B. Kulasekera & Nikolaos Limnios & Bo Henry Lindqvist, 2011. "Applied Nonparametric Statistics in Reliability," Springer Series in Reliability Engineering, Springer, number 978-0-85729-118-9, February.
    6. Jens Perch Nielsen & Carsten Tanggaard, 2001. "Boundary and Bias Correction in Kernel Hazard Estimation," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 28(4), pages 675-698, December.
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