1. Definition

2. Examples

3. General remarks

4. Comparisons with other classes of models

5. See also

6. Notes

7. Bibliography

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, {{math|𝒫}}, is indexed by some set {{math|Θ}}. For each {{math|θ ∈ Θ}}, let {{math|P_θ}} denote the corresponding member of the collection; so {{math|P_θ}} is a cumulative distribution function. Then a statistical model can be written as

The model is a parametric model if {{math|Θ ⊆ ℝ^k}} for some positive integer {{math|k}}.

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 {{math|λ > 0}}:

  

  where {{math|p_λ}} is the probability mass function. This family is an exponential family.

• The normal family is parametrized by {{math|θ {{=}} (μ, σ)}}, where {{math|μ ∈ ℝ}} is a location parameter and {{math|σ > 0}} 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 {{math|θ {{=}} (λ, β, μ)}}:

  

• The binomial model is parametrized by {{math|θ {{=}} (n, p)}}, where {{math|n}} is a non-negative integer and {{math|p}} is a probability (i.e. {{math|p ≥ 0}} and {{math|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 {{math|θ ↦ P_θ}} is invertible, i.e. there are no two different parameter values {{math|θ₁}} and {{math|θ₂}} such that {{math|P_θ1 {{=}} P_θ2}}.

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:{{Citation needed|date=October 2010}}

• 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.^[1] 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.^[2] This difficulty can be avoided by considering only "smooth" parametric models.

See also

• Parametric family
• Parametric statistics
• Statistical model
• Statistical model specification

Notes

Bibliography

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