Parametric model

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 “nonparametric” 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 μ be a fixed σ-finite measure on a probability space (Ω, ℱ), and $\scriptstyle {\mathcal {M}}_{\mu }$ the collection of all probability measures dominated by μ. 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. Θ is an open subset of R^k.
2. The map

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

   from Θ to L²(μ) 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 Θ.
4. The k×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 ∇ƒ_θ.
  2. The score function

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

     belongs to the space L²(P_θ) of square-integrable functions with respect to the measure P_θ.
  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)

References

• Bickel, Peter J. and Doksum, Kjell A. (2001). Mathematical Statistics: Basic and Selected Topics, Volume 1 (Second (updated printing 2007) ed.). Pearson Prentice-Hall.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Efficient and adaptive estimation for semiparametric models. Springer: New York. 1998. ISBN 0-387-98473-9. {{cite book}}: Unknown parameter |authors= ignored (help)CS1 maint: publisher location (link)
• Davidson, A.C. (2003). Statistical Models. Cambridge University Press.
• Freedman, David A. (2009). Statistical Models: Theory and Practice (Second ed.). Cambridge University Press. ISBN 9780521671057. {{cite book}}: External link in |publisher= (help)
• Le Cam, Lucien (2000). Asymptotics in statistics: some basic concepts. Springer. ISBN 0-387-95036-2. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Lehmann, Erich (1983). Theory of Point Estimation.
• Lehmann, Erich (1959). Testing Statistical Hypotheses.
• Liese, Friedrich and Miescke, Klaus-J. (2008). Statistical Decision Theory: Estimation, Testing, and Selection. Springer.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Pfanzagl, Johann; with the assistance of R. Hamböker (1994). Parametric Statistical Theory. Walter de Gruyter. ISBN 3-11-01-3863-8. MR1291393