Parametric model

This article is about statistics. For mathematical and computer representation of objects, see Solid modeling.

In statistics, a parametric model or parametric family or finite-dimensional model is a particular class of statistical model. Specifically, a parametric model is a family of distributions that can be indexed using a finite number of (typically real) parameters. These parameters are usually collected together to form a single k-dimensional parameter vector θ = (θ₁, θ₂, …, 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. 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.

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 . The set of all allowable values for the parameter is denoted Θ ⊆ R^k, and the model itself is written as

${\mathcal {P}}={\big \{}P_{\theta }\ {\big |}\ \theta \in \Theta {\big \}}.$

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

${\mathcal {P}}={\big \{}f_{\theta }\ {\big |}\ \theta \in \Theta {\big \}}.$

The parametric model is called identifiable if the mapping θ ↦ P_θ is invertible, that is there are no two different parameter values θ₁ and θ₂ such that P_θ1 = P_θ2.

Examples

• The Poisson family of distributions is parametrized by a single number λ > 0:

  ${\mathcal {P}}={\Big \{}\ p_{\lambda }(j)={\tfrac {\lambda ^{j}}{j!}}e^{-\lambda },\ j=0,1,2,3,\dots \ {\Big |}\ \lambda >0\ {\Big \}},$

  where p_λ is the probability mass function. This family is an exponential family.
• The normal family is parametrized by θ = (μ, σ), where μ ∈ ℝ is a location parameter, and σ > 0 is a scale parameter:

  ${\mathcal {P}}={\Big \{}\ f_{\theta }(x)={\tfrac {1}{{\sqrt {2\pi }}\sigma }}\exp \left(-{\tfrac {(x-\mu )^{2}}{2\sigma ^{2}}}\right)\ {\Big |}\ \mu \in \mathbb {R} ,\sigma >0\ {\Big \}}.$

  This parametrized family is both an exponential family and a location-scale family.
• The Weibull translation model has three parameters θ = (λ, β, μ):

  ${\mathcal {P}}={\Big \{}\ f_{\theta }(x)={\tfrac {\beta }{\lambda }}\left({\tfrac {x-\mu }{\lambda }}\right)^{\beta -1}\!\exp \!{\big (}\!-\!{\big (}{\tfrac {x-\mu }{\lambda }}{\big )}^{\beta }{\big )}\,\mathbf {1} _{\{x>\mu \}}\ {\Big |}\ \lambda >0,\,\beta >0,\,\mu \in \mathbb {R} \ {\Big \}}.$

See also

• Parametric family
• Parametric statistics
• Statistical model
• Statistical model specification

Bibliography

• Bickel, Peter J.; Doksum, Kjell A. (2001), Mathematical Statistics: Basic and selected topics, vol. Volume 1 (Second (updated printing 2007) ed.), Prentice-Hall {{citation}}: |volume= has extra text (help)
• 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
• Freedman, David A. (2009), Statistical Models: Theory and practice (Second ed.), Cambridge University Press, ISBN 978-0-521-67105-7
• 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
• Lehmann, Erich L.; Romano, Joseph P. (2005), Testing Statistical Hypotheses (3rd 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