Parametric model


In statistics, a parametric model or parametric family or finite-dimensional model is a particular class of statistical models. Specifically, a parametric model is a family of probability distributions that has a finite number of parameters.

Definition

A statistical model is a collection of probability distributions on some sample space. We assume that the collection,, is indexed by some set. The set is called the parameter set or, more commonly, the parameter space. For each, let denote the corresponding member of the collection; so is a cumulative distribution function. Then a statistical model can be written as
The model is a parametric model if for some positive integer.
When the model consists of absolutely continuous distributions, it is often specified in terms of corresponding probability density functions:

Examples

General remarks

A parametric model is called identifiable if the mapping is invertible, i.e. there are no two different parameter values and such that.

Comparisons with other classes of models

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:
Some statisticians believe that the concepts "parametric", "non-parametric", and "semi-parametric" are ambiguous. It can also be noted that the set of all probability measures has cardinality of continuum, and therefore it is possible to parametrize any model at all by a single number in interval. This difficulty can be avoided by considering only "smooth" parametric models.