The Annals of Statistics

The spatial distribution in infinite dimensional spaces and related quantiles and depths

Anirvan Chakraborty and Probal Chaudhuri

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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.

Article information

Ann. Statist., Volume 42, Number 3 (2014), 1203-1231.

First available in Project Euclid: 20 June 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation
Secondary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case) 60G12: General second-order processes

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


Chakraborty, Anirvan; Chaudhuri, Probal. The spatial distribution in infinite dimensional spaces and related quantiles and depths. Ann. Statist. 42 (2014), no. 3, 1203--1231. doi:10.1214/14-AOS1226.

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