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Generalized Quantile Processes Based on Multivariate Depth Functions, with Applications in Nonparametric Multivariate Analysis

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  • Serfling, Robert

Abstract

Statistical depth functions are being used increasingly in nonparametric multivariate data analysis. In a broad treatment of depth-based methods, Liu, Parelius, and Singh ("Multivariate analysis by date depth: Descriptive statistics, graphics and inference (with discussion)," 1999) include several devices for visualizing selected multivariate distributional characteristics by one-dimensional curves constructed in terms of given depth functions. Here we show how these tools may be represented as special depth-based cases of generalized quantile functions introduced by J. H. J. Einmahl and D. M. Mason (1992, Ann. Statist.20, 1062-1078). By specializing results of the latter authors to the depth-based case, we develop an easily applied general result on convergence of sample depth-based generalized quantile processes to a Brownian bridge. As applications, we obtain the asymptotic behavior of sample versions of depth-based curves for "scale" and "kurtosis" introduced by Liu, Parelius and Singh. The kurtosis curve is actually a Lorenz curve designed to measure heaviness of tails of a multivariate distribution. We also obtain the asymptotic distribution of the quantile process of the sample depth values.

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  • 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.
    • Handle: RePEc:eee:jmvana:v:83:y:2002:i:1:p:232-247
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    References listed on IDEAS

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    1. Einmahl, J. H.J. & Mason, D.M., 1992. "Generalized quantile processes," Other publications TiSEM b2a76bac-045d-457f-869f-d, Tilburg University, School of Economics and Management.
    2. Di Bucchianico, A. & Einmahl, J.H.J. & Mushkudiani, N.A., 2001. "Smallest nonparametric tolerance regions," Other publications TiSEM 436f9be2-d0ad-49af-b6df-9, Tilburg University, School of Economics and Management.
    3. Gastwirth, Joseph L, 1971. "A General Definition of the Lorenz Curve," Econometrica, Econometric Society, vol. 39(6), pages 1037-1039, November.
    4. Beirlant, J. & Mason, D. M. & Vynckier, C., 1999. "Goodness-of-fit analysis for multivariate normality based on generalized quantiles," Computational Statistics & Data Analysis, Elsevier, vol. 30(2), pages 119-142, April.
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    Cited by:

    1. Wang, Jin, 2019. "Asymptotics of generalized depth-based spread processes and applications," Journal of Multivariate Analysis, Elsevier, vol. 169(C), pages 363-380.
    2. Romanazzi, Mario, 2009. "Data depth, random simplices and multivariate dispersion," Statistics & Probability Letters, Elsevier, vol. 79(12), pages 1473-1479, June.
    3. Torres, Raúl & Lillo, Rosa E. & Laniado, Henry, 2015. "A directional multivariate value at risk," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 111-123.
    4. Torres Díaz, Raúl Andrés & Michele, Carlo de & Lillo Rodríguez, Rosa Elvira & Laniado Rodas, Henry, 2016. "Directional multivariate extremes in environmental phenomena," DES - Working Papers. Statistics and Econometrics. WS 23419, Universidad Carlos III de Madrid. Departamento de Estadística.
    5. Wang, Jin & Zhou, Weihua, 2012. "A generalized multivariate kurtosis ordering and its applications," Journal of Multivariate Analysis, Elsevier, vol. 107(C), pages 169-180.
    6. Agostinelli, Claudio, 2018. "Local half-region depth for functional data," Journal of Multivariate Analysis, Elsevier, vol. 163(C), pages 67-79.
    7. Wang, Jin & Serfling, Robert, 2006. "Influence functions for a general class of depth-based generalized quantile functions," Journal of Multivariate Analysis, Elsevier, vol. 97(4), pages 810-826, April.
    8. Merlo, Luca & Petrella, Lea & Salvati, Nicola & Tzavidis, Nikos, 2022. "Marginal M-quantile regression for multivariate dependent data," Computational Statistics & Data Analysis, Elsevier, vol. 173(C).
    9. Ra'ul Torres & Rosa E. Lillo & Henry Laniado, 2015. "A Directional Multivariate Value at Risk," Papers 1502.00908, arXiv.org.
    10. 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.

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