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, …, θk).
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.
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A parametric model is a collection of probability distributions such that each member of this collection, Pθ, is described by a finite-dimensional parameter θ. The set of all allowable values for the parameter is denoted Θ ⊆ Rk, and the model itself is written as
When the model consists of absolutely continuous distributions, it is often specified in terms of corresponding probability density functions:
The parametric model is called identifiable if the mapping θ ↦ Pθ is invertible, that is there are no two different parameter values θ1 and θ2 such that Pθ1 = Pθ2.
- The Poisson family of distributions is parametrized by a single number λ > 0:
- 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:
- The Weibull translation model has three parameters θ = (λ, β, μ):
Regular parametric model
Let μ be a fixed σ-finite measure on a probability space (Ω, ℱ), and the collection of all probability measures dominated by μ. Then we will call a regular parametric model if the following requirements are met:
- Θ is an open subset of Rk.
- The map
- The map (defined above) is continuous on Θ.
- The k×k Fisher information matrix
- Sufficient conditions for regularity of a parametric model in terms of ordinary differentiability of the density function ƒθ are following:
- The density function ƒθ(x) is continuously differentiable in θ for μ-almost all x, with gradient ∇ƒθ.
- The score function
- The Fisher information matrix I(θ), defined as
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 -consistent and efficient estimator of its parameter θ.
- Statistical model
- Parametric family
- Parametrization (i.e., coordinate system)
- Parsimony (with regards to the trade-off of many or few parameters in data fitting)
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- Bickel, Peter J.; Klaassen, Chris A.J.; Ritov, Ya’acov; Wellner Jon A. (1998). Efficient and adaptive estimation for semiparametric models. Springer: New York. ISBN 0-387-98473-9.
- 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; Lo Yang, Grace (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 and 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. ISBN 3-11-013863-8. MR 1291393