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Definition. A set of probability measures on indexed by a parameter is said to be a parametric model or parametric family if a only if the parameter space is a subset .
What this definition says is that distributions belonging to a parametric model can be indexed by a finite dimensional parameter. A given parameter corresponds to a single distribution . 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.
- 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
Thus the family of normal distributions is parametrized by . In this case the parameter space is given by .
- For each positive real number λ there is a Poisson distribution whose expected value is λ. Its probability mass function is
Thus the family of Poisson distributions is parametrized by the positive number and the parameter space is given by .
The family of Poisson distributions is an exponential family.
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