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Parametric model
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<h2 id="definition">Definition</h2> <p>A <a href="/facts/Statistical_model/jKkT5ftm">statistical model</a> is a collection of <a href="/facts/Probability_distribution/EpsKKVRu">probability distributions</a> on some <a href="/facts/Sample_space/WVldkN4F">sample space</a>. We assume that the collection, <i>𝒫</i>, is indexed by some set Θ. The set Θ is called the parameter set or, more commonly, the <a href="/facts/Parameter_space/LpdMY5gc">parameter space</a>. For each <i>θ</i> ∈ Θ, let <i>Fθ</i> denote the corresponding member of the collection; so <i>Fθ</i> is a <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a>. Then a statistical model can be written as </p> P = { F θ | θ ∈ Θ } . {\displaystyle {\mathcal {P}}={\big \{}F_{\theta }\ {\big |}\ \theta \in \Theta {\big \}}.} <p>The model is a parametric model if Θ ⊆ ℝ<i>k</i> for some positive integer <i>k</i>. </p><p>When the model consists of absolutely continuous distributions, it is often specified in terms of corresponding <a href="/facts/Probability_density_function/zvfybna4">probability density functions</a>: </p> P = { f θ | θ ∈ Θ } . {\displaystyle {\mathcal {P}}={\big \{}f_{\theta }\ {\big |}\ \theta \in \Theta {\big \}}.} <h2 id="examples">Examples</h2> <ul><li>The <a href="/facts/Poisson_distribution/CvCzXkHr">Poisson family</a> of distributions is parametrized by a single number <i>λ</i> > 0:</li></ul> 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 \}},} <p>where <i>pλ</i> is the <a href="/facts/Probability_mass_function/LhurokRt">probability mass function</a>. This family is an <a href="/facts/Exponential_family/1LkkqEIf">exponential family</a>. </p> <ul><li>The <a href="/facts/Normal_distribution/UapjjPyQ">normal family</a> is parametrized by <i>θ</i> = (<i>μ</i>, <i>σ</i>), where <i>μ</i> ∈ ℝ is a location parameter and <i>σ</i> > 0 is a scale parameter:</li></ul> 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 \}}.} <p>This parametrized family is both an <a href="/facts/Exponential_family/1LkkqEIf">exponential family</a> and a <a href="/facts/Location-scale_family/8o5sgzfC">location-scale family</a>. </p> <ul><li>The <a href="/facts/Weibull_distribution/1fEWxbKl">Weibull translation model</a> has a three-dimensional parameter <i>θ</i> = (<i>λ</i>, <i>β</i>, <i>μ</i>):</li></ul> 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 \}}.} <ul><li>The <a href="/facts/Binomial_distribution/UMoFMjDj">binomial model</a> is parametrized by <i>θ</i> = (<i>n</i>, <i>p</i>), where <i>n</i> is a non-negative integer and <i>p</i> is a probability (i.e. <i>p</i> ≥ 0 and <i>p</i> ≤ 1):</li></ul> 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 \}}.} <p>This example illustrates the definition for a model with some discrete parameters. </p> <h2 id="general-remarks">General remarks</h2> <p>A parametric model is called <a href="/facts/Identifiable/YfE1ydRa">identifiable</a> if the mapping <i>θ</i> ↦ <i>Pθ</i> is invertible, i.e. there are no two different parameter values <i>θ</i>1 and <i>θ</i>2 such that <i>P</i><i>θ</i>1 = <i>P</i><i>θ</i>2. </p> <h2 id="comparisons-with-other-classes-of-models">Comparisons with other classes of models</h2> <p><a href="/facts/Parametric_statistics/Uvq9SJV9">Parametric models</a> are contrasted with the <a href="/facts/Semiparametric_model/L5jTbkSc">semi-parametric</a>, semi-nonparametric, and <a href="/facts/Non-parametric_model/svTDJHTb">non-parametric models</a>, all of which consist of an infinite set of "parameters" for description. The distinction between these four classes is as follows: </p> <ul><li>in a "<i><a href="/facts/Parametric_statistics/Uvq9SJV9">parametric</a></i>" model all the parameters are in finite-dimensional parameter spaces;</li> <li>a model is "<i><a href="/facts/Nonparametric_statistics/svTDJHTb">non-parametric</a></i>" if all the parameters are in infinite-dimensional parameter spaces;</li> <li>a "<i>semi-parametric</i>" model contains finite-dimensional parameters of interest and infinite-dimensional <a href="/facts/Nuisance_parameter/acJWVqxj">nuisance parameters</a>;</li> <li>a "<i>semi-nonparametric</i>" model has both finite-dimensional and infinite-dimensional unknown parameters of interest.</li></ul> <p>Some statisticians believe that the concepts "parametric", "non-parametric", and "semi-parametric" are ambiguous.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> It can also be noted that the set of all probability measures has <a href="/facts/Cardinality/m6VPojwT">cardinality</a> of <a href="/facts/Continuum_(set_theory)/ErGtLklG">continuum</a>, and therefore it is possible to parametrize any model at all by a single number in (0,1) interval.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> This difficulty can be avoided by considering only "smooth" parametric models. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Parametric_family/YoEWw0I8">Parametric family</a></li> <li><a href="/facts/Parametric_statistics/Uvq9SJV9">Parametric statistics</a></li> <li><a href="/facts/Statistical_model/jKkT5ftm">Statistical model</a></li> <li><a href="/facts/Statistical_model_specification/d8MInLmL">Statistical model specification</a></li></ul> <h2 id="notes">Notes</h2> <h2 id="bibliography">Bibliography</h2> <ul><li><a href="/facts/Peter_J._Bickel/sDYbUSpo">Bickel, Peter J.</a>; Doksum, Kjell A. (2001), <i>Mathematical Statistics: Basic and selected topics</i>, vol. 1 (Second (updated printing 2007) ed.), <a href="/facts/Prentice-Hall/bynWAuat">Prentice-Hall</a></li> <li><a href="/facts/Peter_J._Bickel/sDYbUSpo">Bickel, Peter J.</a>; Klaassen, Chris A. J.; Ritov, Ya’acov; Wellner, Jon A. (1998), <i>Efficient and Adaptive Estimation for Semiparametric Models</i>, Springer</li> <li>Davison, A. C. (2003), <i>Statistical Models</i>, <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a></li> <li><a href="/facts/Lucien_Le_Cam/t2DVPly7">Le Cam, Lucien</a>; <a href="/facts/Grace_Yang/AdeKntfN">Yang, Grace Lo</a> (2000), <i>Asymptotics in Statistics: Some basic concepts</i> (2nd ed.), Springer</li> <li><a href="/facts/Erich_Leo_Lehmann/hun7hqhM">Lehmann, Erich L.</a>; <a href="/facts/George_Casella/CTjcaJCt">Casella, George</a> (1998), <i>Theory of Point Estimation</i> (2nd ed.), Springer</li> <li>Liese, Friedrich; Miescke, Klaus-J. (2008), <i>Statistical Decision Theory: Estimation, testing, and selection</i>, Springer</li> <li>Pfanzagl, Johann; with the assistance of R. Hamböker (1994), <i>Parametric Statistical Theory</i>, <a href="/facts/Walter_de_Gruyter/2wJSEv3m">Walter de Gruyter</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1291393">1291393</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Le Cam & Yang 2000, §7.4 - Le Cam, Lucien; Yang, Grace Lo (2000), Asymptotics in Statistics: Some basic concepts (2nd ed.), Springer <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>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 <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>