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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

P = { F θ   |   θ ∈ Θ } . {\displaystyle {\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:

P = { f θ   |   θ ∈ Θ } . {\displaystyle {\mathcal {P}}={\big \{}f_{\theta }\ {\big |}\ \theta \in \Theta {\big \}}.}

Examples

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

P = {   p λ ( j ) = λ j j ! e − λ ,   j = 0 , 1 , 2 , 3 , …   | λ > 0   } , {\displaystyle {\mathcal {P}}={\Big \{}\ p_{\lambda }(j)={\tfrac {\lambda ^{j}}{j!}}e^{-\lambda },\ j=0,1,2,3,\dots \ {\Big |}\;\;\lambda >0\ {\Big \}},}

where pλ 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:

P = {   f θ ( x ) = 1 2 π σ exp ⁡ ( − ( x − μ ) 2 2 σ 2 )   | μ ∈ R , σ > 0   } . {\displaystyle {\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):

P = {   f θ ( x ) = β λ ( x − μ λ ) β − 1 exp ( − ( x − μ λ ) β ) 1 { x > μ }   | λ > 0 , β > 0 , μ ∈ R   } . {\displaystyle {\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 ≥ 0 and p ≤ 1):

P = {   p θ ( k ) = n ! k ! ( n − k ) ! p k ( 1 − p ) n − k ,   k = 0 , 1 , 2 , … , n   | n ∈ Z ≥ 0 , p ≥ 0 ∧ p ≤ 1 } . {\displaystyle {\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 ↦ Pθ is invertible, i.e. there are no two different parameter values θ1 and θ2 such that Pθ1 = Pθ2.

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.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.

See also

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

Notes

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

References

1. Le Cam & Yang 2000, §7.4 - Le Cam, Lucien; Yang, Grace Lo (2000), Asymptotics in Statistics: Some basic concepts (2nd ed.), Springer ↩

2. Bickel et al. 1998, p. 2 - Bickel, Peter J.; Klaassen, Chris A. J.; Ritov, Ya’acov; Wellner, Jon A. (1998), Efficient and Adaptive Estimation for Semiparametric Models, Springer ↩