{{Short description|Type of statistical model}} {{about|statistics|mathematical and computer representation of objects|Solid modeling}} {{Multiple issues| {{Context|date=November 2022}}{{Lead rewrite|date=April 2026}} {{Copy edit|date=April 2026}} }} 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== {{no footnotes|section|date=May 2012}} A statistical model is a collection of probability distributions on some sample space. We assume that the collection, {{math|''𝒫''}}, is indexed by some set {{math|Θ}}. The set {{math|Θ}} is called the '''parameter set''' or, more commonly, the '''parameter space'''. For each {{math|''θ'' ∈ Θ}}, let {{math|''F<sub>θ</sub>''}} denote the corresponding member of the collection; so {{math|''F<sub>θ</sub>''}} is a cumulative distribution function. Then a statistical model can be written as : <math> \mathcal{P} = \big\{ F_\theta\ \big|\ \theta\in\Theta \big\}. </math>
The model is a '''parametric model''' if {{math|Θ ⊆ ℝ<sup>''k''</sup>}} for some positive integer {{math|''k''}}.
When the model consists of absolutely continuous distributions, it is often specified in terms of corresponding probability density functions: : <math> \mathcal{P} = \big\{ f_\theta\ \big|\ \theta\in\Theta \big\}. </math>
==Examples== * The Poisson family of distributions is parametrized by a single number {{math|''λ'' > 0}}: : <math> \mathcal{P} = \Big\{\ p_\lambda(j) = \tfrac{\lambda^j}{j!}e^{-\lambda},\ j=0,1,2,3,\dots \ \Big|\;\; \lambda>0 \ \Big\}, </math> where {{math|''p<sub>λ</sub>''}} is the probability mass function. This family is an exponential family.
* The normal family is parametrized by {{math|''θ'' {{=}} (''μ'', ''σ'')}}, where {{math|''μ'' ∈ ℝ}} is a location parameter and {{math|''σ'' > 0}} is a scale parameter: : <math> \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\}. </math> This parametrized family is both an exponential family and a location-scale family.
* The Weibull translation model has a three-dimensional parameter {{math|''θ'' {{=}} (''λ'', ''β'', ''μ'')}}: : <math> \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\}, </math> where <math>\beta</math> is the shape parameter, <math>\lambda</math> is the scale parameter and <math>\mu</math> is the location parameter.
* The binomial model is parametrized by {{math|''θ'' {{=}} (''n'', ''p'')}}, where {{math|''n''}} is a non-negative integer and {{math|''p''}} is a probability (i.e. {{math|''p'' ≥ 0}} and {{math|''p'' ≤ 1}}): : <math> \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\}. </math> This example illustrates the definition for a model with some discrete parameters.
==General remarks== A parametric model is called identifiable if the mapping {{math|''θ'' ↦ ''P<sub>θ</sub>''}} is invertible, i.e. there are no two different parameter values {{math|''θ''<sub>1</sub>}} and {{math|''θ''<sub>2</sub>}} such that {{math|''P''<sub>''θ''<sub>1</sub></sub> {{=}} ''P''<sub>''θ''<sub>2</sub></sub>}}.
==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:{{Citation needed|date=October 2010}} * 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.<ref>{{harvnb|Le Cam| Yang|2000}}, §7.4</ref> 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.<ref>{{harvnb|Bickel|Klaassen| Ritov| Wellner| 1998|page=2}}</ref> This difficulty can be avoided by considering only "smooth" parametric models.
==See also== * Parametric family * Parametric statistics * Statistical model * Statistical model specification
==Notes== {{Reflist}}
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{{DEFAULTSORT:Parametric Model}} Category:Parametric statistics Category:Statistical models