Parametric model

This article is about statistics. For mathematical and computer representation of objects, see Solid modeling.

In statistics, a parametric model or parametric family or finite-dimensional model is a family of distributions that can be described using a finite number of parameters. These parameters are usually collected together to form a single k-dimensional parameter vector θ = (θ₁, θ₂, …, θ_k).

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. 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. This difficulty can be avoided by considering only "smooth" parametric models.

Definition

A parametric model is a collection of probability distributions such that each member of this collection, P_θ, is described by a finite-dimensional parameter θ. The set of all allowable values for the parameter is denoted Θ ⊆ R^k, and the model itself is written as

${\mathcal {P}}={\big \{}P_{\theta }\ {\big |}\ \theta \in \Theta {\big \}}.$

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

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

The parametric model is called identifiable if the mapping θ ↦ P_θ is invertible, that is there are no two different parameter values θ₁ and θ₂ such that P_θ1 = P_θ2.

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 ∈ R is a location parameter, and σ > 0 is a scale parameter. This parametrized family is both an exponential family and a location-scale family:

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

• The Weibull translation model has three parameters θ = (λ, β, μ):

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

  This model is not regular (see definition below) unless we restrict β to lie in the interval (2, +∞).

Regular parametric model

Let $\mu$ be a fixed σ-finite measure on a measurable space $(\Omega ,{\mathcal {F}})$, and $\scriptstyle {\mathcal {M}}_{\mu }$ the collection of all probability measures dominated by $\mu$. Then we will call ${\mathcal {P}}\!=\!\{P_{\theta }|\,\theta \in \Theta \}\subseteq {\mathcal {M}}_{\mu }$ a regular parametric model if the following requirements are met:

1. $\Theta$ is an open subset of $\mathbb {R} ^{k}$.
2. The map

   $\theta \mapsto s(\theta )={\sqrt {dP_{\theta }/d\mu }}$

   from $\Theta$ to $L^{2}(\mu )$ is Fréchet differentiable: there exists a vector ${\dot {s}}(\theta )=({\dot {s}}_{1}(\theta ),\,\ldots ,\,{\dot {s}}_{k}(\theta ))$ such that

   $\lVert s(\theta +h)-s(\theta )-{\dot {s}}(\theta )'h\rVert =o(|h|){\text{ as }}h\to 0,$

   where ′ denotes matrix transpose.
3. The map $\theta \mapsto {\dot {s}}(\theta )$ (defined above) is continuous on $\Theta$.
4. The $k\times k$ Fisher information matrix

   $I(\theta )=4\int {\dot {s}}(\theta ){\dot {s}}(\theta )'d\mu$

   is non-singular.

Properties

• Sufficient conditions for regularity of a parametric model in terms of ordinary differentiability of the density function ƒ_θ are following:
  1. The density function ƒ_θ(x) is continuously differentiable in θ for μ-almost all $x$, with gradient $\nabla f_{\theta }$.
  2. The score function

     $z_{\theta }={\frac {\nabla f_{\theta }}{f_{\theta }}}\cdot \mathbf {1} _{\{f_{\theta }>0\}}$

     belongs to the space $L^{2}(P_{\theta })$ of square-integrable functions with respect to the measure $P_{\theta }$.
  3. The Fisher information matrix I(θ), defined as

     $I_{\theta }=\int \!z_{\theta }z_{\theta }'\,dP_{\theta }$

     is nonsingular and continuous in θ.

  If conditions (i)−(iii) hold then the parametric model is regular.

• Local asymptotic normality.
• If the regular parametric model is identifiable then there exists a uniformly $\scriptstyle {\sqrt {n}}$-consistent and efficient estimator of its parameter θ.

See also

• Statistical model
• Parametric family
• Parametrization (i.e. coordinate system)
• Parsimony (with regards to the trade-off of many or few parameters in data fitting)

Bibliography

• Bickel, Peter J.; Doksum, Kjell A. (2001). Mathematical Statistics: Basic and Selected Topics. Vol. Volume 1 (Second (updated printing 2007) ed.). Pearson Prentice-Hall. {{cite book}}: |volume= has extra text (help); Unknown parameter |lastauthoramp= ignored (|name-list-style= suggested) (help)
• Bickel, Peter J.; Klaassen, Chris A. J.; Ritov, Ya’acov; Wellner, Jon A. (1998). Efficient and Adaptive Estimation for Semiparametric Models. Springer.
• Davidson, A. C. (2003). Statistical Models. Cambridge University Press.
• Freedman, David A. (2009). Statistical Models: Theory and Practice (Second ed.). Cambridge University Press. ISBN 978-0-521-67105-7.
• Le Cam, Lucien; Yang, Grace Lo (2000). Asymptotics in Statistics: some basic concepts. Springer. ISBN 0-387-95036-2.
• Lehmann, Erich (1983). Theory of Point Estimation.
• Lehmann, Erich (1959). Testing Statistical Hypotheses.
• Liese, Friedrich; Miescke, Klaus-J. (2008). Statistical Decision Theory: Estimation, Testing, and Selection. Springer. {{cite book}}: Unknown parameter |lastauthoramp= ignored (|name-list-style= suggested) (help)
• Pfanzagl, Johann; with the assistance of R. Hamböker (1994). Parametric Statistical Theory. Walter de Gruyter. ISBN 3-11-013863-8. MR 1291393.