Approximate one-sided tolerance limits in random effects model and in some mixed models and comparisons

Abstract

An approximate closed-form one-sided tolerance limit (TL) in a general mixed model is proposed. One-sided TLs for the distribution of observable random variable and for the distribution of unobservable random variable in one-way random model are obtained as special cases from the one for the general mixed model. Applications to a two-way nested random model are also given. The merits of the TLs are evaluated using Monte Carlo simulation and compared with the existing ones. Our comparison studies indicate that the approximate TLs are quite satisfactory for all parameter and sample size configurations, and better than the existing ones in some cases. Approximate confidence intervals for exceedance probabilities in one-way random effects model are also proposed. The procedures are illustrated using three examples.

Keywords:

• exceedance probability
• generalized variable
• Graybill–Wang approximation
• MOVER
• Satterthwaite approximation
• nested model
• survival probability
• variance components