Predicting future failures in generalized order statistics and related models

Quantile regression predictions are considered for generalized order statistics which extend results previously established for record values and order statistics. In order to derive the results, some (univariate and bivariate) distortion representations of generalized order statistics are established. In particular, prediction of the sth generalized order statistic X(s)∗ based on F given a single (generalized) order statistic X(r)∗ with r<s will be addressed. The presentation includes results for both known and unknown baseline distributions. In the latter case, we consider exponential distributions with unknown mean as well as the proportional hazards model. Using unimodality properties of the marginal density functions (pdf) of generalized order statistics, we find a simple representation of the maximum likelihood prediction of X(s)∗ given X(r)∗.