![]() | This article provides insufficient context for those unfamiliar with the subject.(November 2022) |
In statistics, a parametric model or parametric family or finite-dimensional model is a particular class of statistical models. Specifically, a parametric model is a family of probability distributions that has a finite number of parameters.
![]() | This section includes a
list of references,
related reading, or
external links, but its sources remain unclear because it lacks
inline citations. (May 2012) |
A statistical model is a collection of probability distributions on some sample space. We assume that the collection, 𝒫, is indexed by some set Θ. The set Θ is called the parameter set or, more commonly, the parameter space. For each θ ∈ Θ, let Fθ denote the corresponding member of the collection; so Fθ is a cumulative distribution function. Then a statistical model can be written as
The model is a parametric model if Θ ⊆ ℝk for some positive integer k.
When the model consists of absolutely continuous distributions, it is often specified in terms of corresponding probability density functions:
where pλ is the probability mass function. This family is an exponential family.
This parametrized family is both an exponential family and a location-scale family.
This example illustrates the definition for a model with some discrete parameters.
A parametric model is called identifiable if the mapping θ ↦ Pθ is invertible, i.e. there are no two different parameter values θ1 and θ2 such that Pθ1 = Pθ2.
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]
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.
![]() | This article provides insufficient context for those unfamiliar with the subject.(November 2022) |
In statistics, a parametric model or parametric family or finite-dimensional model is a particular class of statistical models. Specifically, a parametric model is a family of probability distributions that has a finite number of parameters.
![]() | This section includes a
list of references,
related reading, or
external links, but its sources remain unclear because it lacks
inline citations. (May 2012) |
A statistical model is a collection of probability distributions on some sample space. We assume that the collection, 𝒫, is indexed by some set Θ. The set Θ is called the parameter set or, more commonly, the parameter space. For each θ ∈ Θ, let Fθ denote the corresponding member of the collection; so Fθ is a cumulative distribution function. Then a statistical model can be written as
The model is a parametric model if Θ ⊆ ℝk for some positive integer k.
When the model consists of absolutely continuous distributions, it is often specified in terms of corresponding probability density functions:
where pλ is the probability mass function. This family is an exponential family.
This parametrized family is both an exponential family and a location-scale family.
This example illustrates the definition for a model with some discrete parameters.
A parametric model is called identifiable if the mapping θ ↦ Pθ is invertible, i.e. there are no two different parameter values θ1 and θ2 such that Pθ1 = Pθ2.
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]
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.