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.
Definition
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
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.
Examples The Poisson family of distributions is parametrized by a single number :
where is the probability mass function. This family is an exponential family.
The normal family is parametrized by , where is a location parameter and is a scale parameter:
This parametrized family is both an exponential family and a location-scale family.
The Weibull translation model has a three-dimensional parameter :
where is the shape parameter, is the scale parameter and is the location parameter.
The binomial model is parametrized by , where is a non-negative integer and is a probability (i.e. and ):
This example illustrates the definition for a model with some discrete parameters.
General remarks A parametric model is called identifiable if the mapping is invertible, i.e. there are no two different parameter values and such that .
Comparisons with other classes of models 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: in a "parametric" model all the parameters are in finite-dimensional parameter spaces; a model is "non-parametric" if all the parameters are in infinite-dimensional parameter spaces; a "semi-parametric" model contains finite-dimensional parameters of interest and infinite-dimensional nuisance parameters; a "semi-nonparametric" model has both finite-dimensional and infinite-dimensional unknown parameters of interest.
Some statisticians believe that the concepts "parametric", "non-parametric", and "semi-parametric" are ambiguous., §7.4 It can also be noted that the set of all probability measures has a cardinality of the continuum, and therefore it is possible to parametrize any model at all by a single number in the (0,1) interval. This difficulty can be avoided by considering only "smooth" parametric models.
See also Parametric family Parametric statistics Statistical model Statistical model specification
Notes
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