Prediction of Gamma failure times

Statistically-independent operating components, each of which follows a Gamma failure-law, are simultaneously put into service. Two predictors of later failure times, based on observations of earlier failures, are proposed and investigated. The predictors are in the form of estimated conditional mean and median of the value being predicted. Unknown parameters of the underlying failure law are estimated by the method of maximum likelihood (ML), and the predictors are constructed using a parametric bootstrap. These conditional median and mean predictors provide a relatively easy method to compute predictors of future Gamma order statistics. Simulation indicates that these predictors are effective except when the shape parameter of the Gamma distribution is small. Generally, the larger the fraction of available data and the closer the value being predicted, the more accurate the predictions (as anticipated). The simulation also detected some difficulty in implementing ML for the gamma based on type-II censored data when the sample ratio of the geometric mean to the arithmetic mean is very close to 1. This problem warrants further study.