Parametric versus nonparametric tolerance regions in detection problems

A major problem in statistical quality control is to detect a change in the distribution of independent sequentially observed random vectors. The case of a Gaussian pre-change distribution has been extensively analyzed. Here we are concerned with the non-normal multivariate case. In this setup it is natural to use tolerance regions as detection tools. These regions are defined in terms of density level sets, which can be estimated in a plug-in fashion. Under a normal mixture model we compare, through a simulation study, the performance of such a detection scheme for two density estimators: a (parametric) normal mixture and a (nonparametric) kernel estimator. The problem of the bandwidth choice for the latter is addressed. We also obtain a result concerning the convergence rates of the error probabilities under a general parametric model. Finally, a real data example is discussed.