Generalized Quantile Processes Based on Multivariate Depth Functions, with Applications in Nonparametric Multivariate Analysis
✍ Scribed by Robert Serfling
- Publisher
- Elsevier Science
- Year
- 2002
- Tongue
- English
- Weight
- 141 KB
- Volume
- 83
- Category
- Article
- ISSN
- 0047-259X
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✦ Synopsis
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.