Nonparametric statistics

Builds on Statistical model Parametric model

Definition

Let $\mathcal{P}$ be a statistical model, a set of probability distributions on a sample space. The model $\mathcal{P}$ is nonparametric if it is not a parametric model. That is, it cannot be indexed in a natural way by a parameter in a subset of $\R^k$ for a fixed $k$ . Nonparametric statistics is the study of inference in nonparametric models. It also covers distribution-free procedures: tests or intervals whose law under the null hypothesis is the same for all distributions in a nonparametric class.

Remarks

"Natural" means that the indexing map has the regularity required of a parametric model. Any set of the size of $\R$ is in bijection with $\R$ , so a bare bijection does not count. Example: the set of all distributions on $\R$ with a continuous distribution function is nonparametric. Example: the sign test of $H_0 \colon$ the median is $0$ is distribution-free. Under $H_0$ , the number of positive observations is binomial with parameters $n$ and $1/2$ , for every continuous law with median $0$ . Non-example: the family $\set{\mathcal{N}(\mu, \sigma^2) : \mu \in \R, \sigma > 0}$ is parametric, with $k = 2$ . Semiparametric models mix both kinds, with a finite-dimensional parameter of interest and an infinite-dimensional nuisance part.

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Tag	`3QB691PFPB`
URI	`m2://3QB691PFPB`
Kind	definition
Writing	W0 stub
MSC	62G05, 62G10
Defines	Nonparametric statistics
Uses	Statistical model, Parametric model

Nonparametric statistics

Definition

Remarks

Reading path

Further reading

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