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:[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[edit]

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[edit]

  • The Poisson family of distributions is parametrized by a single number 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 μR 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[edit]

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:[3]

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

Properties[edit]

  • Sufficient conditions for regularity of a parametric model in terms of ordinary differentiability of the density function ƒθ are following:[4]
    1. The density function ƒθ(x) is continuously differentiable in θ for μ-almost all , with gradient .
    2. The score function
      belongs to the space of square-integrable functions with respect to the measure .
    3. 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.[5]

See also[edit]

Notes[edit]

  1. ^ Le Cam & Yang 2000, §7.4
  2. ^ Bickel et al. 1998, p. 2
  3. ^ Bickel et al. 1998, p. 12
  4. ^ Bickel et al. 1998, p.13, prop.2.1.1
  5. ^ Bickel et al. 1998, Theorems 2.5.1, 2.5.2

Bibliography[edit]