# Parametric vs. non-parametric models

Yao Yao on June 20, 2015

In statistics, a parametric model or parametric family or finite-dimensional model is a family of distributions that can be described using a finite number of parameters. These parameters are usually collected together to form a single $k$-dimensional parameter vector $\theta = (\theta_1, \theta_2, \cdots, \theta_k)$.

Parametric models are contrasted with the semi-parametric, semi-nonparametric, and non-parametric models, all of which consist of an infinite set of “parameters” for description. The distinction between these four classes is as follows:

• a model is “parametric” if all the parameters are in finite-dimensional parameter spaces;
• a model is “non-parametric” if all the parameters are in infinite-dimensional parameter spaces;
• a “semi-parametric” model contains finite-dimensional parameters of interest and infinite-dimensional nuisance ([ˈnju:sns], something annoying) parameters;
• a “semi-nonparametric” model contains finite-dimensional and infinite-dimensional unknown parameters of interest.