Tolerance intervals for symmetric location-scale families based on uncensored or censored samples

A double integration method for generating exact tolerance limit factors for normal populations

This article introduces a new method for generating the exact one-sided and two-sided tolerance limit factors for normal populations. This method does not need to handle the noncentral t-distribution at all, but only needs to do a double integration of a joint probability density function with respect to the two independent variables “s” (standard deviation) and “x” (sample mean). The factors generated by this method are investigated through Monte Carlo simulations and compared with the existing factors. As a result, it is identified that the two-sided tolerance limit factors being currently used in practical applications are inaccurate. For the right understanding, some factors generated by this method are presented in Tables along with a guidance for correct use of them. The AQL (Acceptable Quality Level) is a good, common measure about quality of a product lot which was already produced or will be produced. Therefore, when performing sampling inspection on a given lot using a tolerance limit factor, there is a necessity to know the AQL assigned to the factor. This new double integration method even makes it possible to generate the AQLs corresponding to the one-sided and two-sided tolerance limit factors.

Tolerance intervals in statistical software and robustness under model misspecification

A tolerance interval is a statistical interval that covers at least 100ρ% of the population of interest with a 100(1−α)% confidence, where ρ and α are pre-specified values in (0, 1). In many scientific fields, such as pharmaceutical sciences, manufacturing processes, clinical sciences, and environmental sciences, tolerance intervals are used for statistical inference and quality control. Despite the usefulness of tolerance intervals, the procedures to compute tolerance intervals are not commonly implemented in statistical software packages. This paper aims to provide a comparative study of the computational procedures for tolerance intervals in some commonly used statistical software packages including JMP, Minitab, NCSS, Python, R, and SAS. On the other hand, we also investigate the effect of misspecifying the underlying probability model on the performance of tolerance intervals. We study the performance of tolerance intervals when the assumed distribution is the same as the true underlying distribution and when the assumed distribution is different from the true distribution via a Monte Carlo simulation study. We also propose a robust model selection approach to obtain tolerance intervals that are relatively insensitive to the model misspecification. We show that the proposed robust model selection approach performs well when the underlying distribution is unknown but candidate distributions are available.