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Keywords:

  • nonhomogeneous Poisson process;
  • nonparametric estimation;
  • nonparametric prediction intervals

Summary

We consider the situation that repair times of several identically structured technical systems are observed. As an example of such data we discuss the Boeing air conditioner data, consisting of successive failures of the air conditioning system of each member of a fleet of Boeing jet airplanes. The repairing process is assumed to be performed according to a minimal-repair strategy. This reflects the idea that only those operations are accomplished that are absolutely necessary to restart the system after a failure. The ‘after-repair-state’ of the system is the same as it was shortly before the failure. Clearly, the observed repair times contain valuable information about the repair times of an identically structured system put into operation in the future. Thus, for statistical analysis and prediction, it is certainly favourable to take into account all repair times from each system. The resulting pooled sample is used to construct nonparametric prediction intervals for repair times of a future minimal-repair system. To illustrate our results we apply them to the above-mentioned data set. As expected, the maximum coverage probabilities of prediction intervals based on two samples exceed those based on one sample. We show that the relative gain for a two-sample prediction over a one-sample prediction can be substantial. One of the advantages of the present approach is that it allows nonparametric prediction intervals to be constructed directly. This provides a beneficial alternative to existing nonparametric methods for minimal-repair systems that construct prediction intervals via the asymptotic distribution of quantile estimators. Moreover, the prediction intervals presented here are exact regardless of the sample size.