ELLIPSOIDAL TOLERANCE REGIONS AND SIMULTANEOUS TOLERANCE INTERVALS FOR SOME MULTIVARIATE NORMAL POPULATIONS

A tolerance region for a population is a region computed using a random sample, so that the region will include a specified proportion or more of the population, with a given confidence level. The theory of statistical tolerance regions has undergone vigorous development during the last several years. In particular, the computation of satisfactory tolerance regions for multivariate normal populations and multivariate regression models have been investigated, and approximations have been developed for computing the required tolerance factor. The available literature in the case of multivariate normal populations deal with the derivation of ellipsoidal tolerance regions only. The present research is motivated by several observations: for ellipsoidal tolerance regions, the available numerical methods to compute the tolerance factor are unsatisfactory, and the available approximations are not always accurate. Furthermore, the literature does not address the computation of simultaneous tolerance intervals. For computing the tolerance factor required to obtain an ellipsoidal tolerance region, an approximation coupled with Monte Carlo simulation is investigated. The methodology is developed when the unknown population covariance matrix does not have any specific structure, and when it has the intra-class covariance structure. Numerical results show that this approach results in an accurate tolerance factor. The methodology is extended to the case of a multivariate linear regression model for obtaining an ellipsoidal tolerance region under a fixed set of values of the covariates. The second part of the research deals with the derivation of simultaneous tolerance intervals and simultaneous prediction intervals. This is motivated by the fact that an ellipsoidal tolerance region cannot provide information on the distribution of the individual components of the response vector. The case of a general covariance matrix, and that of the intra-class covariance matrix are separately considered, and the derivation of both one-sided and two-sided tolerance intervals are addressed. In the case of the intra-class covariance structure, the derivation of a tolerance rectangle is also investigated. Numerical results are given to assess the accuracy of the proposed solutions, and illustrative examples and applications are provided