Parametric Statistical 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
: $\mathcal = \big\.$

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:
: $\mathcal = \big\.$

Examples

* The Poisson family of distributions is parametrized by a single number :
: $\mathcal = \Big\,$
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:
: $\mathcal = \Big\.$
This parametrized family is both an exponential family and a location-scale family.

* The Weibull translation model has a three-dimensional parameter :
: $\mathcal = \Big\.$

* The binomial model is parametrized by , where is a non-negative integer and is a probability (i.e. and ):
: $\mathcal = \Big\.$
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 (0,1) interval. This difficulty can be avoided by considering only "smooth" parametric models.

See also

* Parametric family
* Parametric statistics
* Statistical model
* Statistical model specification

Notes

Bibliography

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{{DEFAULTSORT:Parametric Model
Parametric statistics
Statistical models