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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).

Contents

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, 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

    P = { P θ   |   θ ∈ Θ } .

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

    P = { f θ   |   θ ∈ Θ } .

    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.

    Examples

  • The Poisson family of distributions is parametrized by a single number λ > 0: P = {   p λ ( j ) = λ j j ! e − λ ,   j = 0 , 1 , 2 , 3 , …   |   λ > 0   } , where pλ 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: P = {   f θ ( x ) = 1 2 π σ e − 1 2 σ 2 ( x − μ ) 2   |   μ ∈ R , σ > 0   } .
  • The Weibull translation model has three parameters θ = (λ, β, μ): P = {   f θ ( x ) = β λ ( x − μ λ ) β − 1 exp ( − ( x − μ λ ) β ) 1 { x > μ }   |   λ > 0 , β > 0 , μ ∈ R   } . 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 measurable space ( Ω , F ) , and M μ the collection of all probability measures dominated by μ . Then we will call P = { P θ | θ ∈ Θ } ⊆ M μ a regular parametric model if the following requirements are met:

    1. Θ is an open subset of R k .
    2. The mapfrom Θ to L 2 ( μ ) is Fréchet differentiable: there exists a vector s ˙ ( θ ) = ( s ˙ 1 ( θ ) , … , s ˙ k ( θ ) ) such that ∥ s ( θ + h ) − s ( θ ) − s ˙ ( θ ) ′ h ∥ = o ( | h | )  as  h → 0 , where ′ denotes matrix transpose.
    3. The map θ ↦ s ˙ ( θ ) (defined above) is continuous on Θ .
    4. The k × k Fisher information matrix I ( θ ) = 4 ∫ s ˙ ( θ ) s ˙ ( θ ) ′ d μ is non-singular.

    Properties

  • Sufficient conditions for regularity of a parametric model in terms of ordinary differentiability of the density function ƒθ are following:
    1. The density function ƒθ(x) is continuously differentiable in θ for μ-almost all x , with gradient ∇ f θ .
    2. The score functionbelongs to the space L 2 ( P θ ) of square-integrable functions with respect to the measure P θ .
    3. The Fisher information matrix I(θ), defined as I θ = ∫ z θ z θ ′ d P θ 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 n -consistent and efficient estimator of its parameter θ.

    • References

    Parametric model Wikipedia
    (Text) CC BY-SA


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