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Depth functions as measures of representativeness

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  • Ye Dong
  • Stephen Lee

Abstract

Data depth provides a natural means to rank multivariate vectors with respect to an underlying multivariate distribution. Most existing depth functions emphasize a centre-outward ordering of data points, which may not provide a useful geometric representation of certain distributional features, such as multimodality, of concern to some statistical applications. Such inadequacy motivates us to develop a device for ranking data points according to their “representativeness” rather than “centrality” with respect to an underlying distribution of interest. Derived essentially from a choice of goodness-of-fit test statistic, our device calls for a new interpretation of “depth” more akin to the concept of density than location. It copes particularly well with multivariate data exhibiting multimodality. In addition to providing depth values for individual data points, depth functions derived from goodness-of-fit tests also extend naturally to provide depth values for subsets of data points, a concept new to the data-depth literature. Copyright Springer-Verlag Berlin Heidelberg 2014

Suggested Citation

  • Ye Dong & Stephen Lee, 2014. "Depth functions as measures of representativeness," Statistical Papers, Springer, vol. 55(4), pages 1079-1105, November.
    • Handle: RePEc:spr:stpapr:v:55:y:2014:i:4:p:1079-1105
    DOI: 10.1007/s00362-013-0555-5
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    1. repec:eca:wpaper:2013/134611 is not listed on IDEAS
    2. Paul R. Rosenbaum, 2005. "An exact distribution‐free test comparing two multivariate distributions based on adjacency," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(4), pages 515-530, September.
    3. Tatjana Lange & Karl Mosler & Pavlo Mozharovskyi, 2014. "Fast nonparametric classification based on data depth," Statistical Papers, Springer, vol. 55(1), pages 49-69, February.
    4. Zhu, Li-Xing & Fang, Kai-Tai & Bhatti, M Ishaq, 1997. "On Estimated Projection Pursuit-Type Crámer-von Mises Statistics, ," Journal of Multivariate Analysis, Elsevier, vol. 63(1), pages 1-14, October.
    5. Boukili Makhoukhi, Mohammed, 2008. "An approximation for the power function of a non-parametric test of fit," Statistics & Probability Letters, Elsevier, vol. 78(8), pages 1034-1042, June.
    6. Anil K. Ghosh & Probal Chaudhuri, 2005. "On Maximum Depth and Related Classifiers," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(2), pages 327-350, June.
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    Cited by:

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    2. Reza Modarres, 2018. "Multinomial interpoint distances," Statistical Papers, Springer, vol. 59(1), pages 341-360, March.

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