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.
A statistical model is a collection of probability distributions on some sample space. We assume that the collection,, is indexed by some set . The set is called the parameter set or, more commonly, the parameter space. For each, let denote the corresponding member of the collection; so is a cumulative distribution function. Then a statistical model can be written as
l{P}=\{F\theta | \theta\in\Theta\}.
The model is a parametric model if for some positive integer .
When the model consists of absolutely continuous distributions, it is often specified in terms of corresponding probability density functions:
l{P}=\{f\theta | \theta\in\Theta\}.
l{P}=\{ pλ(j)=\tfrac{λj}{j!}e-λ, j=0,1,2,3,... | λ>0 \},
l{P}=\{ f\theta(x)=\tfrac{1}{\sqrt{2\pi}\sigma}\exp\left(-\tfrac{(x-\mu)2}{2\sigma2}\right) | \mu\inR,\sigma>0 \}.
l{P}=\{ f\theta(x)=\tfrac{\beta}{λ}\left(\tfrac{x-\mu}{λ}\right)\beta-1 \exp(-(\tfrac{x-\mu}{λ})\beta) 1\{x>\mu\
l{P}=\{ p\theta(k)=\tfrac{n!}{k!(n-k)!}pk(1-p)n-k, k=0,1,2,...,n | n\inZ\ge,p\ge0\landp\le1\}.
A parametric model is called identifiable if the mapping is invertible, i.e. there are no two different parameter values and such that .
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.[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. This difficulty can be avoided by considering only "smooth" parametric models.