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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.


\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.

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