Generalized quantile processes based on multivariate depth functions, with applications in nonparametric multivariate analysis

[摘要] 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. Eimnahl 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. (C) 2002 Elsevier Science (USA).

[发布日期] 2002-10-01

[关键词] generalized quantile processes;statistical depth functions;multivariate analysis;nonparametric methods;multivariate scatter;scale curve;kurtosis;heavy tails;weak convergence