48 Nonparametric Statistics

48.1 Parametric Inference

Parametric inference is based on a family of known probability distributions governed by a defined parameter space.

The goal is to perform inference (or more generally statistics) on the values of the parameters.

48.2 Nonparametric Inference

Nonparametric inference or modeling can be described in two ways (not mutually exclusive):

1. An inference procedure or model that does not depend on or utilize the parametrized probability distribution from which the data are generated.

2. An inference procedure or model that may have a specific structure or based on a specific formula, but the complexity is adaptive and can grow to arbitrary levels of complexity as the sample size grows.

In All of Nonparametric Statistics, Larry Wasserman says:

… it is difficult to give a precise definition of nonparametric inference…. For the purposes of this book, we will use the phrase nonparametric inference to refer to a set of modern statistical methods that aim to keep the number of underlying assumptions as weak as possible.

He then lists five estimation examples (see Section 1.1): distributions, functionals, densities, regression curves, and Normal means.

48.3 Nonparametric Descriptive Statistics

Almost all of the exploratory data analysis methods we covered in the beginning of the course are nonparametric.

Sometimes the exploratory methods are calibrated by known probability distributions, but they are usually informative regardless of the underlying probability distribution (or lack thereof) of the data.

48.4 Semiparametric Inference

Semiparametric inference or modeling methods contain both parametric and nonparametric components.

An example is $$X_i | \mu_i \sim \mbox{Normal}(\mu_i, 1)$$ and $$\mu_i {\; \stackrel{\text{iid}}{\sim}\;}F$$ for some arbitrary distribution $$F$$.