Working Paper 03-70 Statistics and Econometrics Series 17 December 2003

Departamento de Estadística y Econometría Universidad Carlos III de Madrid

Calle Madrid, 126 28903 Getafe (Spain)

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PARAMETRIC VERSUS NONPARAMETRIC TOLERANCE REGIONS IN

DETECTION PROBLEMS

Amparo Baíllo and Antonio Cuevas* Abstract 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 pre-change 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.

Keywords: Level sets; statistical quality control; density estimates; normal mixtures; false alarm probability. *Baíllo, Statistics and Econometrics Department, University Carlos III de Madrid, C/ Madrid, 126, 28903 Getafe (Madrid), e-mail: abaillo@est-econ.uc3m.es; Cuevas, Mathematics Department, University Autónoma de Madrid, 28049 Madrid, e-mail: antonio.cuevas@uam.es. Research partially supported by Spanish grant BFM2001-0169.