# 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 = (θ1, θ2, …, θk).

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, Pθ, is described by a finite-dimensional parameter θ. The set of all allowable values for the parameter is denoted Θ ⊆ Rk, 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 θPθ is invertible, that is there are no two different parameter values θ1 and θ2 such that Pθ1 = Pθ2.

## 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]

1. is an open subset of .
2. The map

from to is Fréchet differentiable: there exists a vector such that

where ′ denotes matrix transpose.

3. The map (defined above) is continuous on .
4. 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 ƒθ are following:[4]
1. The density function ƒθ(x) is continuously differentiable in θ for μ-almost all , with gradient .
2. The score function

belongs to the space of square-integrable functions with respect to the measure .

3. The Fisher information matrix I(θ), defined as

is nonsingular and continuous in θ.

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

• Local asymptotic normality.
• If the regular parametric model is identifiable then there exists a uniformly -consistent and efficient estimator of its parameter θ.[5]

## Notes

1. LeCam 2000, ch.7.4
2. Bickel 1998, p. 2
3. Bickel 1998, p. 12
4. Bickel 1998, p.13, prop.2.1.1
5. Bickel 1998, Theorems 2.5.1, 2.5.2

## Bibliography

• Bickel, Peter J. & Doksum, Kjell A. (2001). 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
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