Parametric model

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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:{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}

  • 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.[1] 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.[2] This difficulty can be avoided by considering only “smooth” parametric models.

Definition

Template:No footnotes 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 θPθ is invertible, that is there are no two different parameter values θ1 and θ2 such that Pθ1 = Pθ2.

Examples

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:[3]

  1. Θ is an open subset of Rk.
  2. The map
    from Θ to L2(μ) is Fréchet differentiable: there exists a vector such that
    where ′ denotes matrix transpose.
  3. The map (defined above) is continuous on Θ.
  4. The k×k Fisher information matrix
    is non-singular.

Properties

See also

Notes

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  4. Template:Harvnb, p.13, prop.2.1.1
  5. Template:Harvnb, Theorems 2.5.1, 2.5.2

References

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