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

Aurore Delaigle and Peter Hall
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.

1. 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.
2. 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.

• 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.