Computational Statistics

, Volume 21, Issue 3–4, pp 523–536 | Cite as

Parametric versus nonparametric tolerance regions in detection problems

Original Paper
First Online:

Abstract

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.

Keywords

Level sets Quality control Density estimates Normal mixtures 
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References

  1. Aitchison J, Dunsmore IR (1975) Statistical prediction analysis. Cambridge University Press, CambridgeMATHGoogle Scholar
  2. Baíllo A (2003) Total error in a plug-in estimator of level sets. Statist Probab Lett 65:411–417MATHMathSciNetCrossRefGoogle Scholar
  3. Baíllo A, Cuevas A, Justel A (2000) Set estimation and nonparametric detection. Can J Statist 28:765–782MATHGoogle Scholar
  4. Baíllo A, Cuesta-Albertos JA, Cuevas A (2001) Convergence rates in nonparametric estimation of level sets. Statist Probab Lett 53:27–35MATHMathSciNetCrossRefGoogle Scholar
  5. Cappé O (2001) A set of MATLAB/OCTAVE functions for the EM estimation of mixtures and hidden Markov models. Downloadable at http://www.tsi.enst.fr/cappe/h2m/Google Scholar
  6. Chatterjee SK, Patra NK (1980) Asymptotically minimal multivariate tolerance sets. Calcutta Statist Assoc Bull 29:73–93MathSciNetGoogle Scholar
  7. Csörgö M, Horváth L (1997) Limit theorems in change-point analysis. Wiley, New YorkMATHGoogle Scholar
  8. Dempster A, Laird N, Rubin D (1977) Maximum likelihood from incomplete data via the EM algorithm. J Roy Statist Soc B 39:1–38MATHMathSciNetGoogle Scholar
  9. Devroye L, Györfi L (1985) Nonparametric density estimation: the L 1 View. Wiley, New YorkGoogle Scholar
  10. Devroye L, Wise G (1980) Detection of abnormal behavior via nonparametric estimation of the support. SIAM J Appl Math 38:480–488MATHMathSciNetCrossRefGoogle Scholar
  11. Di Bucchianico A, Einmahl JHJ, Mushkudiani N (2001) Smallest nonparametric tolerance regions. Ann Statist 29(5):1320–1343MATHMathSciNetCrossRefGoogle Scholar
  12. Duong T, Hazelton ML (2003) Plug-in bandwidth matrices for bivariate kernel density estimation. J Nonparametr Statist 15:17–30MATHMathSciNetCrossRefGoogle Scholar
  13. Fuchs C, Kenett RS (1987) Multivariate tolerance regions and F-tests. J Quality Tech 19:122–131Google Scholar
  14. Fuchs C, Kenett RS (1998) Multivariate quality control. Theory and Applications. Marcel Dekker, New YorkGoogle Scholar
  15. Gombay E (1994) Testing for change-points with rank and sign statistics. Statist Probab Lett 20:49–55MATHMathSciNetCrossRefGoogle Scholar
  16. Gordon L, Pollak M (1995) A robust surveillance scheme for stochastically ordered alternatives. Ann Statist 23(4):1350–1375MATHMathSciNetGoogle Scholar
  17. Hawkins DM, Olwell DH (1998) Cumulative sum charts and charting for quality improvement. Springer, Berlin Heidelberg New YorkMATHGoogle Scholar
  18. Hotelling H (1947) Multivariate quality control illustrated by air testing of sample bombsights. In: Eisenhart C et al. (eds) Selected techniques of statistical analysis. Mc Graw-Hill, New YorkGoogle Scholar
  19. Huskova M (1998) Multivariate rank statistics processes and change-point analysis. In: Ahmed SE et al. (eds) Applied statistical science, vol III. Nova Science Publishers, CommackGoogle Scholar
  20. Leroux BG (1992) Consistent estimation of a mixing distribution. Ann Statist 20:1350–1360MATHMathSciNetGoogle Scholar
  21. Liu RY, Singh K (1993) A quality index based on data depth and multivariate rank tests. J Amer Statist Assoc 88(421):252–260MATHMathSciNetCrossRefGoogle Scholar
  22. Marron JS (1996) Matlab Smoothing Software. http://www.stat.unc.edu/faculty/marron/marron_ software.htmlGoogle Scholar
  23. McLachlan G, Peel D (2000) Finite mixture models. Wiley, New YorkMATHCrossRefGoogle Scholar
  24. Mushkudiani N (2000) Statistical applications of generalized quantiles: nonparametric tolerance regions and P–P Plots. Eindhoven University of Technology, EindhovenGoogle Scholar
  25. Page ES (1954) Continuous inspection schemes. Biometrika 41:100–115MATHMathSciNetCrossRefGoogle Scholar
  26. Polansky AM (2001) A smooth nonparametric approach to multivariate process capability. Technometrics 43:199–211MathSciNetCrossRefGoogle Scholar
  27. Sheather SJ, Jones MC (1991) A reliable data-based bandwidth selection method for kernel density estimation. J Roy Statist Soc Ser B 53:683–690MATHMathSciNetGoogle Scholar
  28. Shewhart WA (1931) The economic control of the quality of manufactured product. Macmillan, New YorkGoogle Scholar
  29. Simonoff JS (1996) Smoothing methods in statistics. Springer, Berlin Heidelberg New YorkMATHGoogle Scholar
  30. Tsybakov AB (1997) On nonparametric estimation of density level sets. Ann Statist 25:948–969MATHMathSciNetCrossRefGoogle Scholar
  31. Wand MP, Jones MC (1994) Multivariate plug-in bandwidth selection. Comput Statist 9:97–116MATHMathSciNetGoogle Scholar
  32. Yakir B (1996) A lower bound on the ARL to detection of a change with a probability constraint on false alarm. Ann Statist 24:431–435MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Physica-Verlag 2006

Authors and Affiliations

  1. 1.Departamento de EstadísticaUniversidad Carlos III de MadridGetafe (Madrid)Spain
  2. 2.Departamento de MatemáticasUniversidad Autónoma de MadridMadridSpain

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