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Instatistics, aparametric model orparametric family orfinite-dimensional model is a particular class ofstatistical models. Specifically, a parametric model is a family ofprobability distributions that has a finite number of parameters.

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Astatistical model is a collection ofprobability distributions on somesample space. We assume that the collection,𝒫, is indexed by some setΘ. The setΘ is called theparameter set or, more commonly, theparameter space. For eachθ ∈ Θ, letF_θ denote the corresponding member of the collection; soF_θ is acumulative distribution function. Then a statistical model can be written as

${\displaystyle \mathcal{P}=\{F_{\theta }|\theta \in \mathrm{\Theta }\}.}$

The model is aparametric model ifΘ ⊆ ℝ^k for some positive integerk.

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

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

• ThePoisson family of distributions is parametrized by a single numberλ > 0:

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

wherep_λ is theprobability mass function. This family is anexponential family.

• Thenormal family is parametrized byθ = (μ,σ), whereμ ∈ ℝ is a location parameter andσ > 0 is a scale parameter:

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

Thisparametrized family is both anexponential family and alocation-scale family.

• TheWeibull translation model has a three-dimensional parameterθ = (λ,β,μ):

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

where${\displaystyle \beta }$ is theshape parameter,${\displaystyle \lambda }$ is thescale parameter and${\displaystyle \mu }$ is thelocation parameter.

• Thebinomial model is parametrized byθ = (n,p), wheren is a non-negative integer andp is a probability (i.e.p ≥ 0 andp ≤ 1):

${\displaystyle \mathcal{P}=\{p_{\theta }(k)=\frac{n!}{k!(n-k)!}{p}^k(1-p{)}^{n-k},k=0,1,2,\dots ,n|n\in {\mathbb{Z}}_{\ge 0},p\ge 0\wedge p\le 1\}.}$

This example illustrates the definition for a model with some discrete parameters.

A parametric model is calledidentifiable if the mappingθ ↦P_θ is invertible, i.e. there are no two different parameter valuesθ₁ andθ₂ such thatP_θ1 =P_θ2.

Parametric models are contrasted with thesemi-parametric,semi-nonparametric, andnon-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]}

• 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-dimensionalnuisance 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 hascardinality ofcontinuum, 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.

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

• Parametric statistics
• Statistical models

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