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Data depth, random simplices and multivariate dispersion

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  • Romanazzi, Mario

Abstract

Depth functions give information not only on the location but also on the dispersion of probability distributions. The Lebesgue integral of Liu's simplicial depth function is equal to the expected volume of the random simplex whose vertices are p+1 independent observations from the relevant distribution. Oja's volume depth is the Lebesgue integral of a linear transformation of the influence function of simplicial depth. The relation of these results with dispersive orderings of distributions is discussed. Some properties of Mahalanobis' and halfspace depth are illustrated.

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  • Romanazzi, Mario, 2009. "Data depth, random simplices and multivariate dispersion," Statistics & Probability Letters, Elsevier, vol. 79(12), pages 1473-1479, June.
    • Handle: RePEc:eee:stapro:v:79:y:2009:i:12:p:1473-1479
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    References listed on IDEAS

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    1. Serfling, Robert, 2002. "Generalized Quantile Processes Based on Multivariate Depth Functions, with Applications in Nonparametric Multivariate Analysis," Journal of Multivariate Analysis, Elsevier, vol. 83(1), pages 232-247, October.
    2. Zuo, Yijun & Serfling, Robert, 2000. "Nonparametric Notions of Multivariate "Scatter Measure" and "More Scattered" Based on Statistical Depth Functions," Journal of Multivariate Analysis, Elsevier, vol. 75(1), pages 62-78, October.
    3. Oja, Hannu, 1983. "Descriptive statistics for multivariate distributions," Statistics & Probability Letters, Elsevier, vol. 1(6), pages 327-332, October.
    4. Mario Romanazzi, 2008. "A note on simplicial depth function," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 92(3), pages 235-253, August.
    5. Giovagnoli, Alessandra & Wynn, H. P., 1995. "Multivariate dispersion orderings," Statistics & Probability Letters, Elsevier, vol. 22(4), pages 325-332, March.
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    Cited by:

    1. Liesa Denecke & Christine Müller, 2014. "New robust tests for the parameters of the Weibull distribution for complete and censored data," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(5), pages 585-607, July.
    2. Giuseppe Pandolfo & Antonio D’ambrosio, 2023. "Clustering directional data through depth functions," Computational Statistics, Springer, vol. 38(3), pages 1487-1506, September.
    3. Liesa Denecke & Christine Müller, 2014. "Consistency of the likelihood depth estimator for the correlation coefficient," Statistical Papers, Springer, vol. 55(1), pages 3-13, February.

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