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.
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):
This model is not regular (see definition below) unless we restrict β to lie in the interval (2, +∞).
Regular parametric model
Let be a fixed σ-finite measure on a measurable space , and the collection of all probability measures dominated by . Then we will call a regular parametric model if the following requirements are met:
- is an open subset of .
- The map
from to is Fréchet differentiable: there exists a vector such that
where ′ denotes matrix transpose.
- The map (defined above) is continuous on .
- The 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 , with gradient .
- The score function
- The Fisher information matrix I(θ), defined as
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 -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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