Parametric model

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, …, θ_k).

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:

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, P_θ, is described by a finite-dimensional parameter θ. The set of all allowable values for the parameter is denoted Θ ⊆ R^k, 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 ↦ P_θ is invertible, that is there are no two different parameter values θ₁ and θ₂ such that P_θ1 = P_θ2.

Let ${\displaystyle \mu }$ be a fixed σ-finite measure on a measurable space ${\displaystyle (\Omega ,{\mathcal {F}})}$, and ${\displaystyle \scriptstyle {\mathcal {M}}_{\mu }}$ the collection of all probability measures dominated by ${\displaystyle \mu }$. Then we will call ${\displaystyle {\mathcal {P}}\!=\!\{P_{\theta }|\,\theta \in \Theta \}\subseteq {\mathcal {M}}_{\mu }}$ a regular parametric model if the following requirements are met:

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