## Doctoral Theses

### Nonparametric Methods for Data in Infinite Dimensional Space.

12-22-2014

12-22-2015

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

#### Supervisor

Chaudhuri, Probal (TSMU-Kolkata; ISI)

#### Abstract (Summary of the Work)

For univariate as well as finite dimensional multivariate data, there is an extensive literature on nonparametric statistical methods. One of the reasons for the popularity of nonparametric methods is that it is often difficult to justify the assumptions (e.g., Gaussian distribution of the data) made in the models used in parametric methods. Nonparametric procedures use more flexible models, which involve less assumptions. So, they are more robust against possible departures from the model assumptions, and are applicable to a wide variety of data. Nonparametric methods outperform their parametric competitors in many situations, where the assumptions required for the parametric methods are not satisfied. Nowadays, with the advancement in the technology and measurement apparatus, statisticians often have to analyze data, which are curves or functions observed over a domain. Such data are increasingly becoming common in various fields of science like biomedical sciences (ECG and EEG curves of patients observed over a time period, MRI and other image data obtained from patients), cognitive sciences (data on hand-writing and speech patterns of subjects), chemical science (spectrometric data observed over a range of wavelengths), environmental science (air pollutant levels at different places over a period of time), meteorology (temperature curves at different places over a year, precipitation levels at different locations during a year) etc (see Ramsay and Silverman (2005) for a detailed exposition). A major difference between this type of data and standard multivariate data is that the set of points in the domain, where one sample observation is recorded may be different than those for the other sample observations. Further, the number of such points is often very large compared to the number of sam- ples making the dimension of the data larger than the sample size. As a result, standard multivariate techniques cannot be used for analyzing such data. However, this type of data can be conveniently handled by viewing them as observations from some infinite dimensional space, e.g., the space of functions defined on an interval in the real line. Due to the advantages in using nonparametric methods for multivariate data lying in finite dimensional spaces, one may expect that such procedures will also be useful for analyzing data, which lie in infinite dimensional spaces. In this thesis, we will address this issue by investigating various nonparametric procedures for such data. Ranks, distributions and quantiles have been used to develop various nonparamet- rie procedures for univariate data (see, e.g., Lehmann (1975) and HÃ¡jek et al. (1999)). Various extensions of these notions are available in the literature for multivariate data Iying in finite dimensional spaces, and several well-known nonparametric procedures have been developed based on them (see, e.g., Puri and Sen (1971) and Oja (2010)). My thesis will be mainly devoted to investigating similar notions in infinite dimensional spaces and studying nonparametric statistical methods based on them. Perhaps, the most popular nonparametric test for univariate data is the Wilcoxon- Mann-Whitney rank sum test for two sample problems. For finite dimensional multi- variate data, several extensions of the Wilcoxon-Mann-Whitney test have been studied, and some of these extensions have been shown to be asymptotically more efficient than the Hotelling's T2 test for many non-Gaussian distributions (see, e.g., Puri and Sen (1971), Randles and Peters (1990), Liu and Singh (1993), Choi and Marden (1997), Chakraborty and Chaudhuri (1999) and Oja (1999)).

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843412

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