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A Depth Function and a Scale Curve Based on Spatial Quantiles

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Statistical Data Analysis Based on the L1-Norm and Related Methods

Part of the book series: Statistics for Industry and Technology ((SIT))

Abstract

Spatial quantiles, based on the L 1 norm in a certain sense, provide an appealing vector extension of univariate quantiles and generate a useful “volume” functional based on spatial “central regions” of increasing size. A plot of this functional as a “spatial scale curve” provides a convenient two-dimensional characterization of the spread of a multivariate distribution of any dimension. We discuss this curve and establish weak convergence of the empirical version. As a tool, we introduce and study a new statistical depth function which is naturally associated with spatial quantiles. Other depth functions that generate L 1-based multivariate quantiles are also noted.

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References

  1. Abdous, B. and Theodorescu, R. (1992). Note on the spatial quantile of a random vector. Statistics and Probability Letters 13 333–336.

    Article  MATH  MathSciNet  Google Scholar 

  2. Avérous, J. and Meste, M. (1990). Location, skewness and tailweight in L s-sense: a coherent approach. Mathematische Operationsforshung und Statistik, Series Statistics 21 57–74.

    MATH  Google Scholar 

  3. Avérous, J. and Meste, M. (1997). Median balls: an extension of the interquantile intervals to multivariate distributions. Journal of Multivariate Analysis 63 222–241.

    Article  MATH  MathSciNet  Google Scholar 

  4. Balanda, K. P. and MacGillivray, H. L. (1990). Kurtosis and spread. The Canadian Journal of Statistics 18 17–30.

    Article  MATH  MathSciNet  Google Scholar 

  5. Chakraborty, B. (2001). On affine equivariant multivariate quantiles. Annals of the Institute of Statistical Mathematics 53 380–403.

    Article  MATH  MathSciNet  Google Scholar 

  6. Chaudhuri, P. (1996). On a geometric notion of quantiles for multivariate data. Journal of the American Statistical Association 91 862–872.

    Article  MATH  MathSciNet  Google Scholar 

  7. Choi, K. and Marden, J. (1997). An approach to multivariate rank tests in multivariate analysis of variance. Journal of the American Statistical Association 92 1581–1590.

    Article  MATH  MathSciNet  Google Scholar 

  8. Donoho, D. L. (1982). Breakdown properties of multivariate location estimators. Ph.D. qualifying paper, Department of Statistics, Harvard University.

    Google Scholar 

  9. Donoho, D. L. and Gasko, M. (1992). Breakdown properties of location estinates based on halfspace depth and projected outlyingness. Annals of Statistics 20 1803–1827.

    Article  MATH  MathSciNet  Google Scholar 

  10. Ferguson, T. S. (1967). Mathematical Statistics: A Decision Theoretic Approach. Academic Press, New York.

    MATH  Google Scholar 

  11. Jan, S. L. and Randles, R. H. (1994). A multivariate signed sum test for the one sample location problem. Journal of Nonparametric Statistics 4 49–63.

    Article  MATH  MathSciNet  Google Scholar 

  12. Kemperman, J. H. B. (1987). The median of a finite measure on a Banach space. In Statistical Data Analysis Based On the L 1-Norm and Related Methods (Y. Dodge, ed.), pp. 217-230, North-Holland.

    Google Scholar 

  13. Koenker, R. and Bassett, G. (1978). Regression quantiles. Journal of the American Statistical Association 82 851–857.

    MathSciNet  Google Scholar 

  14. Koltchinskii, V. (1997). M-estimation, convexity and quantiles. Annals of Statistics 25 435–477.

    Article  MATH  MathSciNet  Google Scholar 

  15. Liu, R. Y. (1990). On a notion of data depth based on random simplices. Annals of Statistics 18 405–414.

    Article  MATH  MathSciNet  Google Scholar 

  16. Liu, R. Y., Parelius, J. M. and Singh, K. (1999). Multivariate analysis by data depth: Descriptive statistics, graphics and inference (with discussion). Annals of Statistics 27 783–858.

    MATH  MathSciNet  Google Scholar 

  17. Liu, R. Y. and Singh, K. (1993). A quality index based on data depth and multivariate rank tests. Journal of the American Statistical Association 88 252–260.

    MATH  MathSciNet  Google Scholar 

  18. Lopuhaä, H. P. and Rousseeuw, P. J. (1991). Breakdown points of affine equivariant estimators of multivariate location and covariance matrices. Annals of Statistics 19 229–248.

    Article  MATH  MathSciNet  Google Scholar 

  19. Marden, J. I. (1998). Bivariate QQ-plots and spider web plots. Statistica Sinica 8 813–826. Sankhya B 36 115-128.

    MATH  MathSciNet  Google Scholar 

  20. Möttönen, J. and Oja, H. (1995). Multivariate spatial sign and rank methods. Journal of Nonparametric Statistics 5 201–213.

    Article  MATH  MathSciNet  Google Scholar 

  21. Möttönen, J., Oja, H., and Tienari, J. (1997). On the efficiency of multivariate spatial sign and rank tests. Annals of Statistics 25 542–552.

    Article  MATH  MathSciNet  Google Scholar 

  22. Rousseeuw, P. J. and Ruts, I. (1999). The depth function of a population distribution. Metrika 49 213–244.

    MATH  MathSciNet  Google Scholar 

  23. Serfling, R. (2002a). Generalized quantile processes based on multivariate depth functions, with applications in nonparametric multivariate analysis. Journal of Multivariate Analysis, in press.

    Google Scholar 

  24. Serfling, R. (2002b). Quantile functions for multivariate analysis: approaches and applications. Statistica Neerlandica 56 214–232.

    Article  MATH  MathSciNet  Google Scholar 

  25. Serfling, R. (2002c). Nonparametric multivariate descriptive measures based on spatial quantiles. Submitted.

    Google Scholar 

  26. Serfling, R. and Wang, J. (2002). Weak convergence of the studentized sample scale curve. In preparation.

    Google Scholar 

  27. Tukey, J. W. (1975). Mathematics and the picturing of data. In Proceedings of the International Congress of Mathematicians, Vancouver 1974 (R. D. James, ed.), 2 523–531.

    MathSciNet  Google Scholar 

  28. Van Keilegom, I. and Hettmansperger, T. (2002). Inference based on multivariate M-estimators based on bivariate censored data. Journal of the American Statistical Association 97 328–336.

    Article  MATH  MathSciNet  Google Scholar 

  29. Zuo, Y. and Serfling, R. (2000a). General notions of statistical depth function. Annals of Statistics 28 461–482.

    Article  MATH  MathSciNet  Google Scholar 

  30. Zuo, Y. and Serfling, R. (2000b). Structural properties and convergence results for contours of sample statistical depth functions. Annals of Statistics 28 483–499.

    Article  MATH  MathSciNet  Google Scholar 

  31. Zuo, Y. and Serfling, R. (2000c). On the performance of some robust nonparametric location measures relative to a general notion of multivariate symmetry. Journal of Statistical Planning and Inference 84 55–79.

    Article  MATH  MathSciNet  Google Scholar 

  32. Zuo, Y. and Serfling, R. (2000d). Nonparametric notions of multivariate “scatter measure” and “more scattered” based on statistical depth functions. Journal of Multivariate Analysis 75 62–78.

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Texas at Dallas, Richardson, Texas, 75083-0688, USA

    Robert Serfling

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  1. Robert Serfling
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Editor information

Editors and Affiliations

  1. Statistics Group, University of Neuchâtel, P.O. Box 805, CH-2002, Neuchâtel, Switzerland

    Yadolah Dodge (Prof. of Statistics and Operation Research) (Prof. of Statistics and Operation Research)

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© 2002 Springer Basel AG

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Serfling, R. (2002). A Depth Function and a Scale Curve Based on Spatial Quantiles. In: Dodge, Y. (eds) Statistical Data Analysis Based on the L1-Norm and Related Methods. Statistics for Industry and Technology. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8201-9_3

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  • DOI: https://doi.org/10.1007/978-3-0348-8201-9_3

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9472-2

  • Online ISBN: 978-3-0348-8201-9

  • eBook Packages: Springer Book Archive

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