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.
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:
where is the probability mass function. This family is an exponential family.
This parametrized family is both an exponential family and a location-scale family.
where is the shape parameter, is the scale parameter and is the location parameter.
This example illustrates the definition for a model with some discrete parameters.
A parametric model is called identifiable if the mapping is invertible, i.e. there are no two different parameter values and such that .
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 (0,1) interval. This difficulty can be avoided by considering only "smooth" parametric models.