Parametric model

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.

Definition

A statistical model is a collection of probability distributions on some sample space. We assume that the collection,

P

{\displaystyle {\mathcal {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

P

=

{

F

θ

|

θ

∈

Θ

}

.

{\displaystyle {\mathcal {P}}={\big \{}F_{\theta }\ {\big |}\ \theta \in \Theta {\big \}}.}

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:

P

=

{

f

θ

|

θ

∈

Θ

}

.

{\displaystyle {\mathcal {P}}={\big \{}f_{\theta }\ {\big |}\ \theta \in \Theta {\big \}}.}

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

}

,

{\displaystyle {\mathcal {P}}={\Big \{}\ p_{\lambda }(j)={\tfrac {\lambda ^{j}}{j!}}e^{-\lambda },\ j=0,1,2,3,\dots \ {\Big |}\;\;\lambda >0\ {\Big \}},}

where pλ is the probability mass function.