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June 2014 The spatial distribution in infinite dimensional spaces and related quantiles and depths
Anirvan Chakraborty, Probal Chaudhuri
Ann. Statist. 42(3): 1203-1231 (June 2014). DOI: 10.1214/14-AOS1226


The spatial distribution has been widely used to develop various nonparametric procedures for finite dimensional multivariate data. In this paper, we investigate the concept of spatial distribution for data in infinite dimensional Banach spaces. Many technical difficulties are encountered in such spaces that are primarily due to the noncompactness of the closed unit ball. In this work, we prove some Glivenko–Cantelli and Donsker-type results for the empirical spatial distribution process in infinite dimensional spaces. The spatial quantiles in such spaces can be obtained by inverting the spatial distribution function. A Bahadur-type asymptotic linear representation and the associated weak convergence results for the sample spatial quantiles in infinite dimensional spaces are derived. A study of the asymptotic efficiency of the sample spatial median relative to the sample mean is carried out for some standard probability distributions in function spaces. The spatial distribution can be used to define the spatial depth in infinite dimensional Banach spaces, and we study the asymptotic properties of the empirical spatial depth in such spaces. We also demonstrate the spatial quantiles and the spatial depth using some real and simulated functional data.


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Anirvan Chakraborty. Probal Chaudhuri. "The spatial distribution in infinite dimensional spaces and related quantiles and depths." Ann. Statist. 42 (3) 1203 - 1231, June 2014.


Published: June 2014
First available in Project Euclid: 20 June 2014

zbMATH: 1305.62141
MathSciNet: MR3224286
Digital Object Identifier: 10.1214/14-AOS1226

Primary: 62G05
Secondary: 60B12 , 60G12

Keywords: Asymptotic relative efficiency , Bahadur representation , DD-plot , Donsker property , Gâteaux derivative , Glivenko–Cantelli property , Karhunen–Loève expansion , smooth Banach space

Rights: Copyright © 2014 Institute of Mathematical Statistics


Vol.42 • No. 3 • June 2014
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