10 Nonparametric Regression

This chapter surveys regression techniques that relax functional-form assumptions. Beginning with kernel and local-polynomial estimators, we derive bias-variance trade-offs and bandwidth-selection criteria. We then explore splines, generalized additive models, regression trees, random forests, and wavelet regression, emphasizing their interpretability and robustness. Multivariate nonparametrics are introduced through radial-basis functions. Confidence-interval construction via asymptotics and bootstrap methods is detailed, and a forward-looking conclusion discusses how nonparametric ideas underpin modern machine-learning algorithms, reinforcing the evolving landscape of regression analysis.

Nonparametric regression refers to a class of regression techniques that do not assume a specific functional form (e.g., linear, polynomial of fixed degree) for the relationship between a predictor \(x \in \mathbb{R}\) (or \(\mathbf{x} \in \mathbb{R}^p\)) and a response variable \(y \in \mathbb{R}\). Instead, nonparametric methods aim to estimate this relationship directly from the data, allowing the data to “speak for themselves.”

In a standard regression framework, we have a response variable \(Y\) and one or more predictors \(\mathbf{X} = (X_1, X_2, \ldots, X_p)\). Let us start with a univariate setting for simplicity. We assume the following model:

\[ Y = m(x) + \varepsilon, \]

where:

• \(m(x) = \mathbb{E}[Y \mid X = x]\) is the regression function we aim to estimate,
• \(\varepsilon\) is a random error term (noise) with \(\mathbb{E}[\varepsilon \mid X = x] = 0\) and constant variance \(\operatorname{Var}(\varepsilon) = \sigma^2\).

In parametric regression (e.g., Linear Regression), we might assume \(m(x)\) has a specific form, such as:

\[ m(x) = \beta_0 + \beta_1 x + \cdots + \beta_d x^d, \]

where \(\beta_0, \beta_1, \ldots, \beta_d\) are parameters to be estimated. In contrast, nonparametric regression relaxes this assumption and employs methods that can adapt to potentially complex shapes in \(m(x)\) without pre-specifying its structure.

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