One-sided simultaneous prediction limits for left-censored normal random variables

The presence of left censoring in environmental and engineering applications, complicates tests of hypotheses and interval estimation. We develop two new prediction limit formulae for at least p of m future observations from r locations, based on n background measurements from a left-censored normal distribution. The first method ignores uncertainty in the parameter estimates (MUPL), whereas the second method (MAUPL) incorporates this uncertainty. The two new methods are compared in theory and via simulation with the two traditional approaches to this problem. The first and most widely used method involves simple imputation of one-half of the censoring point (e.g., a detection limit) and computes the mean and variance of the distribution and the prediction limit as if all data were observed (SIUPL). The second method uses the MLE of the mean and variance of the censored normal distribution in the prediction limit formula for the complete data case (IUPL). Results of our simulation study reveal that the simple method (SIUPL) provides grossly inflated Type I error rates, where the second traditional method (IUPL) performs well except when the sample size is small (e.g., n = 8) and the censoring proportion is large (e.g., 50%). The MAUPL method is overly conservative, but appropriate for those cases in which we re-quire that the Type I error rate never exceeds the nominal level. The MUPL method appears to outperform all other methods in terms of achieving the intended nominal level and maximizing statistical power.