Abstract
In this paper, we present an accurate procedure, called “within-sample prediction of order statistics,” to obtain prediction limits for the number of failures that will be observed in a future inspection of a sample of units, based only on the results of the first in-service inspection of the same sample. The failure-time of such units is modeled with a two-parameter Weibull distribution indexed by scale and shape parameters β and δ, respectively. It will be noted that in the literature only the case is considered when the scale parameter β is unknown, but the shape parameter δ is known. As a rule, in practice the Weibull shape parameter δ is not known. Instead it is estimated subjectively or from relevant data. Thus, its value is uncertain. This δ uncertainty may contribute greater uncertainty to the construction of prediction limits for a future number of failures. In this paper, we consider the case when both parameters β and δ are unknown. The technique proposed here for constructing prediction limits emphasizes pivotal quantities relevant for obtaining ancillary statistics and represents a special case of the method of invariant embedding of sample statistics into a performance index applicable whenever the statistical problem is invariant under a group of transformations, which acts transitively on the parameter space. Application to other distributions could follow directly.
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Acknowledgment
This research was supported in part by Grant No. 06.1936, Grant No. 07.2036, Grant No. 09.1014, and Grant No. 09.1544 from the Latvian Council of Science and the National Institute of Mathematics and Informatics of Latvia.
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Nechval, N.A., Nechval, K.N., Purgailis, M. (2013). Weibull Prediction Limits for a Future Number of Failures Under Parametric Uncertainty. In: Ao, SI., Gelman, L. (eds) Electrical Engineering and Intelligent Systems. Lecture Notes in Electrical Engineering, vol 130. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-2317-1_23
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