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.

Robust Model Selection Criteria Based on Pseudodistances

In this paper, we introduce a new class of robust model selection criteria. These criteria are defined by estimators of the expected overall discrepancy using pseudodistances and the minimum pseudodistance principle. Theoretical properties of these criteria are proved, namely asymptotic unbiasedness, robustness, consistency, as well as the limit laws. The case of the linear regression models is studied and a specific pseudodistance based criterion is proposed. Monte Carlo simulations and applications for real data are presented in order to exemplify the performance of the new methodology. These examples show that the new selection criterion for regression models is a good competitor of some well known criteria and may have superior performance, especially in the case of small and contaminated samples.