Minimum Volume Sets in Statistics: Recent Developments

  • Wolfgang Polonik
Part of the Studies in Classification, Data Analysis, and Knowledge Organization book series (STUDIES CLASS)

Summary

We present a survey of recent developments in statistics connected to minimum volume (MV)-sets. MV-sets are sets which carry high mass concentration. Such sets are of interest in cluster analysis and can for example be used for estimating level sets of densities. Further statistical applications are prediction, testing for multimodality, goodness of fit, data analysis, regression problems, and density estimation.

Keywords

Estima 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. ANDREWS, D. W.; BICKEL, P. J.; HAMPEL, F. R.; HUBER, P. J.; RODGERS, W. H. and TUKEY, J. W. (1972): Robust estimation of location: survey and advances. Princeton Univ. Press, Princeton, N.J.Google Scholar
  2. BARBE, P. and WEI, S. (1994). A test for multimodality. Unpubl. Manuscript.Google Scholar
  3. BARLOW, R. E.; BARTHOLOMEW, D. J.; BREMNER, J. M., and BRUNK, H. D.(1972): Statistical inference under order restrictions. Wiley, London.Google Scholar
  4. BEIRLANT, J. and EINMAHL, J. H. J. (1995). Maximal type test statistics based on conditional processes. Statist. Neerlandica, 49, 1–8.CrossRefGoogle Scholar
  5. BEIRLANT, J. and EINMAHL, J. H. J. (1996). Asymptotic confidence intervals for the length of the shortt under random censoring. To appear in J. Statist. Planning and Inference.Google Scholar
  6. CHERNOFF, H. (1964): Estimation of the mode. Ann. Inst. Statist. Math. 16, 31–41.CrossRefGoogle Scholar
  7. DAVIES, P. L. (1992): The asymptotics of Rousseeuws minimum volume ellipsoid estimator. Ann. Statist., 20, 1828–1843.CrossRefGoogle Scholar
  8. DALENIUS, T. (1965): The mode — a neglected statistical parameter. J. Roy. Statist. Soc. A, 128, 110–117.CrossRefGoogle Scholar
  9. EINMAHL, J. H. J. and MASON, D. M. (1992). Generalized quantite processes. Ann. Statist., 20, 1062–1078.CrossRefGoogle Scholar
  10. GRENANDER, U. (1956): On the theory of mortality measurement, Part II. Skand. Akt., 39, 125–153.Google Scholar
  11. GRÜBEL, R. (1988): The length of the shorth. Ann. Statist, 16, 619–628.CrossRefGoogle Scholar
  12. HARTIGAN, J. A. (1975): Clustering algorithms. Wiley, New York.Google Scholar
  13. HARTIGAN, J. A. (1987): Estimation of a convex density contour in two dimensions. J. Amer. Statist. Assoc., 82, 267–270.CrossRefGoogle Scholar
  14. LIENTZ, B. P. (1970): Results on non-parametric modal intervals. SIAM J. Appl. Math., 19, 356–366.CrossRefGoogle Scholar
  15. MÜLLER, D. W. and SAWITZKI, G. (1987): Using excess mass estimates to investigate the modality of a distribution. Preprint Nr. 398, SFB 123, Univ. Heidelberg.Google Scholar
  16. MÜLLER, D. W. and SAWITZKI, G. (1992): Excess mass estimates and tests of multimodality. J. Amer. Statist. Assoc., 86, 738–746.Google Scholar
  17. NOLAN, D. (1991): The excess mass ellipsoid. J. Multivarite Anal., 39, 348–371.CrossRefGoogle Scholar
  18. POLONIK, W. (1994): Minimum volume sets and generalized quantile processes. Beiträge zur Statistik Nr. 20, Institut für Angewandte Mathematik, Universität Heidelberg.Google Scholar
  19. POLONIK, W. (1995a): Measuring mass concentrations and estimating density contour clusters — an excess mass approach. Ann. Statist., 23, 855–881.CrossRefGoogle Scholar
  20. POLONIK, W. (1995b): Density estimation under qualitative assumptions in higher dimensions. J. Multivariate Anal., 55, 61–81.CrossRefGoogle Scholar
  21. POLONIK, W. (1995c): The silhouette, concentration functions, and ML-density estimation under order restrictions. Technical Report No. 445, Department of Statistics, University of California, Berkeley.Google Scholar
  22. POLONIK, W. (1996): Testing for concentration and goodness-of-fit in higher dimensions-(asymptotically) distribution-free methods. Beiträge zur Statistik Nr. 33, Institut für Angewandte Mathematik, Universität Heidelberg.Google Scholar
  23. ROUSSEEUW, P. J. (1986): Multivariate estimation with high breakdown point. In: W. Grossmann et al. (eds): Mathematical statistics and applications, Reidel, Dordrecht, 283–297.Google Scholar
  24. SAWITZKI, G. (1994). Diagnostic plots for one-dimensional data. In: Dirschedl P. and Ostermann, R. (eds.): Papers collected at the occasion of the 25th Conference on Statistical Computing at Schloss Reisensburg. Physica, Heidelberg.Google Scholar
  25. VENTER, J. H. (1967). On estimation of the mode. Ann. Math. Statist., 38, 1446–1455.CrossRefGoogle Scholar
  26. YAO, Q. and TONG, H. (1994). Quantifiying the inference of initial values on nonlinear prediction. J. Roy. Statist. Soc. B, 56, 701–725.Google Scholar
  27. YAO, Q. (1995). Conditional predictive regions for stochastic processes. Unpubl. manuscript.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Wolfgang Polonik
    • 1
  1. 1.Institut für Angewandte MathematikUniversität HeidelbergHeidelbergGermany

Personalised recommendations