Parametric model

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Type of statistical model

title: "Parametric model" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["parametric-statistics", "statistical-models"] description: "Type of statistical model" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Parametric_model" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Type of 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, 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 Fθ denote the corresponding member of the collection; so Fθ is a cumulative distribution function. Then a statistical model can be written as : \mathcal{P} = \big{ F_\theta\ \big|\ \theta\in\Theta \big}.

The model is a parametric model if Θ ⊆ ℝk for some positive integer k.

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}.

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|;; \lambda0 \ \Big}, where pλ is the probability mass function. This family is an exponential family.

  • The normal family is parametrized by , 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}, \sigma0 \ \Big}. This parametrized family is both an exponential family and a location-scale family.

  • The Weibull translation model has a three-dimensional parameter : : \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|;; \lambda0,, \beta0,, \mu\in\mathbb{R} \ \Big}, where \beta is the shape parameter, \lambda is the scale parameter and \mu is the location parameter.

  • The binomial model is parametrized by , where n is a non-negative integer and p is a probability (i.e. p ≥ 0 and p ≤ 1): : \mathcal{P} = \Big{\ p_\theta(k) = \tfrac{n!}{k!(n-k)!}, p^k (1-p)^{n-k},\ k=0,1,2,\dots, n \ \Big|;; n\in\mathbb{Z}_{\ge 0},, p \ge 0 \land p \le 1\Big}. This example illustrates the definition for a model with some discrete parameters.

General remarks

A parametric model is called identifiable if the mapping θ ↦ Pθ is invertible, i.e. there are no two different parameter values θ1 and θ2 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.

Notes

Bibliography

  • {{citation |author1-last = Bickel | author1-first = Peter J. | author1-link = Peter J. Bickel |author2-last = Doksum | author2-first = Kjell A. | title = Mathematical Statistics: Basic and selected topics | volume = 1 | edition = Second (updated printing 2007) | year = 2001 | publisher = Prentice-Hall
  • {{citation | author1-last = Bickel | author1-first = Peter J. | author1-link = Peter J. Bickel | author2-last = Klaassen | author2-first = Chris A. J. | author3-last = Ritov | author3-first = Ya’acov | author4-first = Jon A. | author4-last = Wellner | year = 1998 | title= Efficient and Adaptive Estimation for Semiparametric Models | publisher = Springer
  • {{citation | last = Davison | first = A. C. | title = Statistical Models | publisher = Cambridge University Press | year = 2003
  • {{citation | author-last = Le Cam | author-first = Lucien | author-link = Lucien Le Cam | author2-last = Yang | author2-first = Grace Lo | author2-link = Grace Yang | title = Asymptotics in Statistics: Some basic concepts | edition = 2nd | year = 2000 | publisher = Springer
  • {{citation | author1-last = Lehmann | author1-first = Erich L. | author1-link = Erich Leo Lehmann | author2-last = Casella | author2-first = George | author2-link = George Casella | title = Theory of Point Estimation | edition = 2nd | year = 1998 | publisher = Springer
  • {{citation | author1-last = Liese| author1-first = Friedrich | author2-last = Miescke| author2-first = Klaus-J. | title = Statistical Decision Theory: Estimation, testing, and selection | year = 2008 | publisher = Springer
  • {{citation | title = Parametric Statistical Theory | last1 = Pfanzagl | first1 = Johann | last2 = with the assistance of R. Hamböker | year = 1994 | publisher = Walter de Gruyter |mr=1291393}}

References

  1. {{harvnb. Le Cam. Yang. 2000, §7.4
  2. {{harvnb. Bickel. Klaassen. Ritov. Wellner. 1998

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