# Parametric model

In statistics, a **parametric model** or **parametric family** or **finite-dimensional model** is a family of distributions that can be described using a finite number of parameters. These parameters are usually collected together to form a single *k*-dimensional *parameter vector* *θ* = (*θ*_{1}, *θ*_{2}, …, *θ*_{k}).

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.

## Definition

A **parametric model** is a collection of probability distributions such that each member of this collection, *P _{θ}*, is described by a finite-dimensional parameter

*θ*. The set of all allowable values for the parameter is denoted Θ ⊆

**R**

^{k}, and the model itself is written as

When the model consists of absolutely continuous distributions, it is often specified in terms of corresponding probability density functions:

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

*θ*

_{1}and

*θ*

_{2}such that

*P*

_{θ1}=

*P*

_{θ2}.

### 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 three parameters
*θ*= (*λ*,*β*,*μ*):This model is

**not**regular (see definition below) unless we restrict*β*to lie in the interval (2, +∞).

## Regular parametric model

Let be a fixed σ-finite measure on a measurable space , and the collection of all probability measures dominated by . Then we will call a **regular parametric model** if the following requirements are met:^{[3]}

- is an open subset of .
- The map
from to is Fréchet differentiable: there exists a vector such that

where ′ denotes matrix transpose.

- The map (defined above) is continuous on .
- The Fisher information matrix
is non-singular.

### Properties

- Sufficient conditions for regularity of a parametric model in terms of ordinary differentiability of the density function ƒ
_{θ}are following:^{[4]}- The density function ƒ
_{θ}(*x*) is continuously differentiable in*θ*for*μ*-almost all , with gradient . - The score function
belongs to the space of square-integrable functions with respect to the measure .

- The Fisher information matrix
*I*(*θ*), defined asis nonsingular and continuous in

*θ*.

If conditions (i)−(iii) hold then the parametric model is regular.

- The density function ƒ
- Local asymptotic normality.
- If the regular parametric model is identifiable then there exists a uniformly -consistent and efficient estimator of its parameter
*θ*.^{[5]}

## See also

- Statistical model
- Parametric family
- Parametrization (i.e., coordinate system)
- Parsimony (with regards to the trade-off of many or few parameters in data fitting)
- Parametricism

## Notes

- ↑ LeCam 2000, ch.7.4
- ↑ Bickel 1998, p. 2
- ↑ Bickel 1998, p. 12
- ↑ Bickel 1998, p.13, prop.2.1.1
- ↑ Bickel 1998, Theorems 2.5.1, 2.5.2

## Bibliography

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*Mathematical Statistics: Basic and Selected Topics*. Volume 1 (Second (updated printing 2007) ed.). Pearson Prentice-Hall. - Bickel, Peter J.; Klaassen, Chris A.J.; Ritov, Ya’acov; Wellner Jon A. (1998).
*Efficient and adaptive estimation for semiparametric models*. Springer: New York. ISBN 0-387-98473-9. - Davidson, A.C. (2003).
*Statistical Models*. Cambridge University Press. - Freedman, David A. (2009).
*Statistical Models: Theory and Practice*(Second ed.). Cambridge University Press. ISBN 978-0-521-67105-7. - Le Cam, Lucien; Lo Yang, Grace (2000).
*Asymptotics in statistics: some basic concepts*. Springer. ISBN 0-387-95036-2. - Lehmann, Erich (1983).
*Theory of Point Estimation*. - Lehmann, Erich (1959).
*Testing Statistical Hypotheses*. - 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. ISBN 3-11-013863-8. MR 1291393