This reprint differs from the original in pagination and typographic detail. 1 2 A. CHAKRABORTY AND P. CHAUDHURIare curves or functions and can be modeled as random observations from probability distributions in infinite dimensional spaces. The ECG curves of patients observed over a period of time, the spectrometry curves recorded for a range of wavelengths, the annual temperature curves of different places, etc., are examples of such data. Many of the function spaces, where such data lie, are infinite dimensional Banach spaces. However, many of the wellknown multivariate medians like the simplicial depth median (see [23]), and the simplicial volume median (see [28]) do not have meaningful extensions into such spaces. On the other hand, the spatial median as well as the spatial quantiles extend easily into infinite dimensional Banach spaces (see [13,19] and [34]). The author of [18] proposed functional principal components using the sample spatial median and used those to analyze a data involving the movements of the lips. The authors of [8] considered an updation based estimator of the spatial median, and used it to compute the profile of a typical television audience in France throughout a single day. The spatial median has also been used in [12] to calculate the median profile for the electricity load data in France. Recently, the authors of [17] studied some direction-based quantiles for probability distributions in infinite dimensional Hilbert spaces. These quantiles are defined for unit direction vectors in such spaces, and they extend the finite dimensional quantiles considered in [22]. The principle quantile directions derived from these quantiles were used in [17] to detect outliers in a dataset of annual age-specific mortality rates of French males between the years 1899 and 2005. The purpose of this article is to investigate the spatial distribution in infinite dimensional Banach spaces, and study their properties along with the spatial quantiles and the spatial depth. There are several mathematical difficulties in dealing with the probability distributions in such spaces. These are primarily due to the noncompactness of the closed unit ball in such spaces. In Section 2, we prove some Glivenko-Cantelli and Donsker-type results for the empirical spatial distribution process arising from data lying in infinite dimensional spaces. In Section 3, we investigate the spatial quantiles in infinite dimensional spaces. A Bahadur-type linear representation of the sample spatial quantiles and their asymptotic Gaussianity are derived. We also study the asymptotic efficiency of the sample spatial median relative to the sample mean for some well-known probability distributions in function spaces. In Section 4, we investigate the spatial depth and its asymptotic properties in infinite dimensional spaces. We also demonstrate how exploratory data analytic tools like the depth-depth plot (DD-plot) (see [24]) can be developed for data in infinite dimensional spaces using the spatial depth. The proofs of the theorems are g...

The concept of data depth leads to a center-outward ordering of multivariate data, and it has been effectively used for developing various data analytic tools. While different notions of depth were originally developed for finite dimensional data, there have been some recent attempts to develop depth functions for data in infinite dimensional spaces. In this paper, we consider some notions of depth in infinite dimensional spaces and study their properties under various stochastic models. Our analysis shows that some of the depth functions available in the literature have degenerate behaviour for some commonly used probability distributions in infinite dimensional spaces of sequences and functions. As a consequence, they are not very useful for the analysis of data satisfying such infinite dimensional probability models. However, some modified versions of those depth functions as well as an infinite dimensional extension of the spatial depth do not suffer from such degeneracy, and can be conveniently used for analyzing infinite dimensional data.

Tests based on sample mean vectors and sample spatial signs have been studied in the recent literature for high dimensional data with the dimension larger than the sample size. For suitable sequences of alternatives, we show that the powers of the mean based tests and the tests based on spatial signs and ranks tend to be same as the data dimension grows to infinity for any sample size, when the coordinate variables satisfy appropriate mixing conditions. Further, their limiting powers do not depend on the heaviness of the tails of the distributions. This is in striking contrast to the asymptotic results obtained in the classical multivariate setup. On the other hand, we show that in the presence of stronger dependence among the coordinate variables, the spatial sign and rank based tests for high dimensional data can be asymptotically more powerful than the mean based tests if in addition to the data dimension, the sample size also grows to infinity. The sizes of some mean based tests for high dimensional data studied in the recent literature are observed to be significantly different from their nominal levels. This is due to the inadequacy of the asymptotic approximations used for the distributions of those test statistics. However, our asymptotic approximations for the tests based on spatial signs and ranks are observed to work well when the tests are applied on a variety of simulated and real datasets.

The Wilcoxon-Mann-Whitney test is a robust competitor of the t-test in the univariate setting. For finite dimensional multivariate data, several extensions of the Wilcoxon-Mann-Whitney test have been shown to have better performance than Hotelling's T 2 test for many non-Gaussian distributions of the data. In this paper, we study a Wilcoxon-Mann-Whitney type test based on spatial ranks for data in infinite dimensional spaces. We demonstrate the performance of this test using some real and simulated datasets. We also investigate the asymptotic properties of the proposed test and compare the test with a wide range of competing tests.

In this paper we have suggested normative indices of relative deprivation.The Yitzhaki (1979) index of relative deprivation is a special case of one of our indices.