# 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, *𝒫*, is indexed by some set Θ. The set Θ is called the **parameter set** or, more commonly, the **parameter space**. For each *θ* ∈ Θ, let *P _{θ}* denote the corresponding member of the collection; so

*P*is a cumulative distribution function. Then a statistical model can be written as

_{θ}The model is a **parametric model** if Θ ⊆ ℝ^{k} for some positive integer *k*.

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
*λ*> 0:where

*p*is the probability mass function. This family is an exponential family._{λ} - The normal family is parametrized by
*θ*= (*μ*,*σ*), where*μ*∈ ℝ is a location parameter and*σ*> 0 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
*θ*= (*λ*,*β*,*μ*): - The binomial model is parametrized by
*θ*= (*n*,*p*), where*n*is a non-negative integer and*p*is a probability (i.e.*p*≥ 0 and*p*≤ 1):This example illustrates the definition for a model with some discrete parameters.

## General remarks

A parametric model is called identifiable if the mapping *θ* ↦ *P _{θ}* is invertible, i.e. there are no two different parameter values

*θ*

_{1}and

*θ*

_{2}such that

*P*

_{θ1}=

*P*

_{θ2}.

## 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.[1] 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.[2] This difficulty can be avoided by considering only "smooth" parametric models.

## Notes

- Le Cam & Yang 2000, §7.4
- Bickel et al. 1998, p. 2

## Bibliography

- Bickel, Peter J.; Doksum, Kjell A. (2001),
*Mathematical Statistics: Basic and selected topics*, Volume 1 (Second (updated printing 2007) ed.), Prentice-Hall - Bickel, Peter J.; Klaassen, Chris A. J.; Ritov, Ya’acov; Wellner, Jon A. (1998),
*Efficient and Adaptive Estimation for Semiparametric Models*, Springer - Davison, A. C. (2003),
*Statistical Models*, Cambridge University Press - Le Cam, Lucien; Yang, Grace Lo (2000),
*Asymptotics in Statistics: Some basic concepts*, Springer - Lehmann, Erich L.; Casella, George (1998),
*Theory of Point Estimation*(2nd ed.), Springer - Liese, Friedrich; Miescke, Klaus-J. (2008),
*Statistical Decision Theory: Estimation, testing, and selection*, Springer - Pfanzagl, Johann; with the assistance of R. Hamböker (1994),
*Parametric Statistical Theory*, Walter de Gruyter, MR 1291393