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

The Poisson family of distributions is parametrized by a single number :

where is the probability mass function. This family is an exponential family.

The normal family is parametrized by, where is a location parameter and is a scale parameter:

This parametrized family is both an exponential family and a location-scale family.

The Weibull translation model has a three-dimensional parameter :

where is the shape parameter, is the scale parameter and is the location parameter.

The binomial model is parametrized by, where is a non-negative integer and is a probability :

This example illustrates the definition for a model with some discrete parameters.

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:

in a "parametric" model 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 parameters;
a "semi-nonparametric" model has both finite-dimensional and infinite-dimensional unknown parameters of interest.

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.