Parametric model

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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 μ and each positive number σ² there is a normal distribution whose expected value is μ and whose variance is σ². 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 λ there is a Poisson distribution whose expected value is λ. 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.