Parametric model

Definition. A set $$ \mathcal{P} = \{ P_\theta \mid \theta \in \Theta \} $$ of probability measures $$ P_\theta $$ on $$ (\Omega, \mathcal{F}) $$ indexed by a parameter $$ \theta $$ is said to be a parametric model or parametric family if a only if the parameter space $$ \Omega $$ is a subset $$ \mathbf{R}^n $$.

What this definition says is that distributions belonging to a parametric model can be indexed by a finite dimensional parameter. A given parameter $$ \theta $$ corresponds to a single distribution $$ f_{\theta} $$. If the distributions belonging to a model cannot be indexed by a finite dimensional parameter, then the model is said to be a nonparametric one. A nonparametric model typically consists of a set of unspecified distributions, e.g. continuous distributions.

Models for which the parameter space can be expressed as the Cartesian product of a finite dimensional Euclidean space and an infinite dimensional parameter space are sometimes called semiparametric.

Examples

 * For each real number &mu; and each positive number &sigma;2 there is a normal distribution whose expected value is &mu; and whose variance is &sigma;2. Its probability density function is


 * $$\varphi_{\mu,\sigma^2}(x) = {1 \over \sigma}\cdot{1 \over \sqrt{2\pi}} \exp\left( {-1 \over 2} \left({x - \mu \over \sigma}\right)^2\right)$$

Thus the family of normal distributions is parametrized by $$ \theta = (\mu, \sigma^2) $$. In this case the parameter space $$ \Omega $$ is given by $$ \Omega = \{ (\mu, \sigma^2) \mid \mu \in \mathbf{R}, \, \sigma^2 > 0 \} $$.

This parametrized family is both an exponential family and a location-scale family


 * For each positive real number &lambda; there is a Poisson distribution whose expected value is &lambda;. Its probability mass function is


 * $$f(x) = {\lambda^x e^{-\lambda} \over x!}\ \mathrm{for}\ x\in\{\,0,1,2,3,\dots\,\}.$$

Thus the family of Poisson distributions is parametrized by the positive number $$ \lambda $$ and the parameter space is given by $$ \Omega = \{ \lambda \mid \lambda > 0 \} $$.

The family of Poisson distributions is an exponential family.