Parametric 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 $θ \in Θ$ , let $F θ$ denote the corresponding member of the collection; so $F θ$ 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 $Θ \subseteq ℝ k$ 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 |}\;\;\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 $μ \in ℝ$ 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 a three-dimensional parameter $θ = (λ, β, μ)$ :

{\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 \}}.

The binomial model is parametrized by $θ = (n, p)$ , where $n$ is a non-negative integer and $p$ is a probability (i.e. $p \geq 0$ and $p \leq 1$ ):

{\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} _{\geq 0},\,p\geq 0\land p\leq 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 $θ \mapsto P θ$ is invertible, i.e. there are no two different parameter values $θ 1$ and $θ 2$ such that $P θ 1 = P θ 2$ .

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]}

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.

Notes

^ Le Cam & Yang 2000, §7.4
^ Bickel et al. 1998, p. 2

Bibliography

Bickel, Peter J.; Doksum, Kjell A. (2001), Mathematical Statistics: Basic and selected topics, vol. 1 (Second (updated printing 2007) ed.), Prentice-Hall
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
Le Cam, Lucien; Yang, Grace Lo (2000), Asymptotics in Statistics: Some basic concepts (2nd ed.), Springer
Lehmann, Erich L.; Casella, George (1998), Theory of Point Estimation (2nd 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

[1] Le Cam & Yang 2000, §7.4

[2] Bickel et al. 1998, p. 2

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