One-Sided Approximate Prediction Intervals for at Least p of m Observations From a Gamma Population at Each of r Locations

Abstract

We develop simultaneous approximate statistical prediction limits for a gamma-distributed random variable. Specifically, we develop an upper prediction limit (UPL) for p of m future samples at each of r locations, based on a previous sample of n measurements. A typical example is the environmental monitoring problem in which the distribution of an analyte of concern is typically non-Gaussian, simultaneous determinations are required for multiple locations (e.g., ground-water monitoring wells), and, in the event of an initial exceedance of the prediction limit, one or more verification samples are obtained to confirm evidence of an impact on the environment. For example, consider a ground-water monitoring program with r wells and the requirement that at least p = 1 of the m = 2 samples in each of the r wells be below the UPL. We provide derivation of simultaneous approximate gamma UPLs, illustration of the relevance of the gamma distribution to environmental data, a limited simulation study of type I and II error rates achieved using the method and comparison with normal and nonparametric alternatives, tables that aid computation, and an example using ground-water monitoring data.