Parametric versus nonparametric tolerance regions indetection problems

A major problem in statistical quality control is to detect a change in the underlying distribution of independent sequentially observed random vectors. The case where the prechange distribution is Gaussian has been extensively analyzed. We are concerned here with the less usual non-normal multivariate case. The use of tolerance regions, defined in terms of density level sets, as detection tools arises as a natural choice in this general setup. The required level sets can be estimated in an obvious plug-in fashion, using either nonparametric or (when a parametric model is assumed) parametric density estimators. A result concerning the convergence rates of the error probabilities under a parametric model is obtained. Also, the performance of parametric and non-parametric methods is compared through a simulation study. Finally, a real data example is discussed. In general terms, we conclude that whereas the parametric estimates are, in theory, preferable when the corresponding model holds, the practical difficulties associated with their implementation make non-parametric methods a very reliable and flexible alternative.