# Methodology and theory for the bootstrap; Introduction to nonparametric regression and functional data

**Aurore Delaigle and Peter Hall**

The University of Melbourne

The University of Melbourne

#### Synopsis

##### Part I: Methodology and theory for the bootstrap

The course will begin by discussing the motivation and intuition behind bootstrap methods, and treat a variety of different approaches, including the double bootstrap. Mathematical arguments based on Edgeworth expansions will then be introduced to assess the performance of the bootstrap, and in particular to compare different approaches. The course will also address at least some of the following topics: bootstrap methods for time series, the m-out-of-n bootstrap, and bootstrap methods for nonparametric curve estimation.

##### Part II: Nonparametric regression and introduction to functional data analysis

In the second part of the course we will introduce techniques for analysing data that are in the form of curves, such as, for example, yearly rainfall or temperature curves, growth curves, etc. We will start by introducing nonparametric smoothing techniques such as kernel and spline methods, which can be used to obtain curves from discretely sampled data. Then, we will consider some of the tools that can be used to describe a sample of random curves (e.g. mean, variance, functional principal component analysis). We will also consider the problems of regression and classification based on functional observations.

- Nonparametric regression: kernel estimators, local polynomial estimators, spline estimators: introduction of methods, discussion of some of their properties. How to compute them in practice + implementation in R. If time permits: derivation of some basic theoretical properties of kernel estimators.
- Functional data: how to obtain functional data; summary statistics (mean, covariance, principal component analysis): methods, description of some theoretical properties, practical computation; introduction to linear regression and nonparametric regression; introduction to classification. If time permits: discussion of some other topics.

#### Contact hours

28 hours, including some computer lab sessions.

#### Prerequisites

Calculus to at least second year university level, including an introduction to Fourier transforms; and at least a second year course in probability theory.

#### Assessment

By take home exam.

#### Resources

Lecture notes for part I and II are provided in hardcopy.

Recommended reading:

- Ferraty and Vieu (2006),
*Nonparametric Functional Data Analysis: Theory And Practice*, Springer. - Hall (1991),
*The Bootstrap and Edgeworth Expansion*, Springer. - Ramsay and Silverman (2005),
*Functional Data Analysis 2nd Edition*, Springer. - Ruppert, Wand and Carroll (2003),
*Semiparametric regression*, Cambridge University Press. - Wand and Jones (1995),
*Kernel smoothing*, Chapman and Hall.

#### About Peter Hall

Peter Hall was born in Sydney, Australia, and received his BSc degree from the University of Sydney in 1974. His MSc and DPhil degrees are from the Australian National University and the University of Oxford, both in 1976. He taught at the University of Melbourne before taking, in 1978, a position at the Australian National University. In November 2006 he moved back to the University of Melbourne. His research interests range across several topics in probability and statistics.

#### About Aurore Delaigle

Aurore Delaigle received her PhD in Statistics from the Université catholique de Louvain in Belgium. She then became a postdoc at the University of California at Davis, and later an Assistant Professor at the University of California in San Diego. Three years after moving to California, she became lecturer and then reader at the University Bristol, UK, where she spent a few years. Aurore is currently a principal researcher and QEII fellow at the University of Melbourne. Her research focuses mainly on nonparametric statistics, functional data analysis and problems of measurement errors.