Parametric Statistical Model Family

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A Parametric Statistical Model Family is a statistical model family that is a parametric model (which assumes a distribution in the sample space described by a low-order vector).




    • 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 “nonparametric” 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.

    • 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 : [math]\displaystyle{ \mathcal{P} = \big\{ P_\theta\ \big|\ \theta\in\Theta \big\}. }[/math] When the model consists of absolutely continuous distributions, it is often specified in terms of corresponding probability density functions: : [math]\displaystyle{ \mathcal{P} = \big\{ f_\theta\ \big|\ \theta\in\Theta \big\}. }[/math] 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.