Tolerance intervals under some discrete models, zero-inflated models and mixture models

Computation of tolerance limits is investigated under the logistic regression model with fixed effects and mixed effects, zero-inflated models, and some mixture models. Under the logistic regression model with fixed effects for binary responses, point-wise and simultaneous tolerance intervals are derived for the number of positive responses among a specified number of Bernoulli trials. Under the logistic regression model with mixed effects, point-wise tolerance limits are derived. The problems were motivated by a ballistic application dealing with testing the ballistic resistance of personal body armor. Methodologies used for the derivation of tolerance limits under the logistic regression model include small sample asymptotics and the bootstrap. Numerical results are reported to assess the accuracy of the proposed methodologies, and they are also illustrated using examples on the ballistic response of armor plates, where it is known that the ballistic response depends on covariates such as the velocity of the projectile. A second set of problems investigated include the computation of upper tolerance limits under zero-inflated log-normal and zero-inflated gamma distributions. Models without covariates and with covariates are separately considered, and point-wise upper tolerance limits are derived under models with covariates. For the zero-inflated log-normal model, techniques used to calculate the tolerance limits include generalized pivotal quantities (GPQs), fiducial quantities, the bootstrap, and the delta method with a bootstrap calibration. Numerical results indicate that the GPQ methodology and the bootstrap calibration of the delta method provide very satisfactory performance. For the zero-inflated gamma model, tolerance limits are calculated based on bootstrapping a pivot-like statistic, and based on the bootstrap calibration of the delta method. Once again, numerical results indicate satisfactory performance. The results are applied to a variety of zero-inflated data sets: health expenditure data, rainfall data, and body armor protection data. The final topic investigated is the derivation of tolerance limits under some mixture models. The normal-normal and lognormal-normal mixture models are first explored. Models with three and four mixtures of normal distributions and the normal-normal regression model are then investigated. Tolerance limits under all of these mixture models are obtained based on a bootstrap calibration of the delta method. This approach produced good results for all the models examined. The case of two mixtures was motivated by an application dealing with the evaluation of the safety of nuclear power plants based on data that are outputs of computer codes used to simulate nuclear accidents. Traditionally, non-parametric tolerance limits are used for this purpose, but the data clearly indicated a mixture distribution: normal-normal or lognormal-normal. Upper tolerance limits computed under the mixture model turned out to be smaller compared to the non-parametric upper tolerance limit. Extensive numerical results and a variety of examples are also included