Tolerance intervals for unbalanced one-way random effects models with covariates and heterogeneous variances

Abstract

Tolerance intervals are useful in practice to help determine limits for detection monitoring or assessment monitoring of factors that may impact the environment, ecological systems, or other biological processes. This article provides a procedure for construction of one-sided and two-sided tolerance intervals for a normally distributed random variable when the mean and variance of its distribution are estimated using data following an unbalanced one-way random effects model with covariates under heterogeneous error variances. The procedure developed here is based on the concept of a generalized pivotal quantity which has been frequently used to obtain confidence intervals in situations where conventional methods are difficult to apply or fail to provide s satisfactory solutions. For the one-sided case, the generalized pivotal quantity approach yields an exact solution. On the other hand, the method leads to good approximate intervals for the two-sided case. This is confirmed by a detailed simulation study, showing that the method may be recommended for practical use. Two real-data examples are given to illustrate the applicability of the proposed procedures.