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, 𝒫 , is indexed by some set Θ . The set Θ is called the parameter set or, more commonly, the parameter space. For each θ ∈ Θ , let Fθ denote the corresponding member of the collection; so Fθ is a cumulative distribution function. Then a statistical model can be written as The model is a parametric model if Θ ⊆ ℝk for some positive integer k . When the model consists of absolutely continuous distributions, it is often specified in terms of corresponding probability density functions:

Examples

λ > 0 where pλ is the probability mass function. This family is an exponential family. , where μ ∈ ℝ is a location parameter and σ > 0 is a scale parameter: This parametrized family is both an exponential family and a location-scale family. , where n is a non-negative integer and p is a probability (i.e. p ≥ 0 and p ≤ 1 ): This example illustrates the definition for a model with some discrete parameters.

General remarks

A parametric model is called identifiable if the mapping θ ↦ Pθ is invertible, i.e. there are no two different parameter values θ1 and θ2 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: Some statisticians believe that the concepts "parametric", "non-parametric", and "semi-parametric" are ambiguous. It can also be noted that the set of all probability measures has cardinality of continuum, and therefore it is possible to parametrize any model at all by a single number in (0,1) interval. This difficulty can be avoided by considering only "smooth" parametric models.