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

Robert Serfling

Journal of Multivariate Analysis, 2002, vol. 83, issue 1, 232-247

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.

Keywords: generalized; quantile; processes; statistical; depth; functions; multivariate; analysis; nonparametric; methods; multivariate; scatter; "scale; curve"; kurtosis; heavy; tails; weak; convergence (search for similar items in EconPapers)
Date: 2002
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Citations: View citations in EconPapers (10)

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