Parametric model

Template:Short description 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 F_θ denote the corresponding member of the collection; so F_θ is a cumulative distribution function. Then a statistical model can be written as

${𝒫}=\left\{F_{\theta }\right|\theta \in \mathrm{\Theta }\}.$

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

When the model consists of absolutely continuous distributions, it is often specified in terms of corresponding probability density functions:

${𝒫}=\left\{f_{\theta }\right|\theta \in \mathrm{\Theta }\}.$

Examples

• The Poisson family of distributions is parametrized by a single number λ > 0:

${𝒫}=\{p_{\lambda }(j)=\frac{{\lambda }^j}{j!}{e}^{-\lambda },j=0,1,2,3,\dots |\lambda >0\},$

where p_λ is the probability mass function. This family is an exponential family.

• The normal family is parametrized by θ = (μ, σ), where μ ∈ ℝ is a location parameter and σ > 0 is a scale parameter:

${𝒫}=\{f_{\theta }(x)=\frac{1}{\sqrt{2\pi }\sigma }\exp (-\frac{(x-\mu {)}^2}{2{\sigma }^2})|\mu \in \mathbb{ℝ},\sigma >0\}.$

This parametrized family is both an exponential family and a location-scale family.

• The Weibull translation model has a three-dimensional parameter θ = (λ, β, μ):

${𝒫}=\{f_{\theta }(x)=\frac{\beta }{\lambda }{\left(\frac{x-\mu }{\lambda }\right)}^{\beta -1}\exp (-\left(\frac{x-\mu }{\lambda }{)}^{\beta }\right){\mathbf{𝟏}}_{\{x>\mu \}}|\lambda >0,\beta >0,\mu \in \mathbb{ℝ}\}.$

• The binomial model is parametrized by θ = (n, p), where n is a non-negative integer and p is a probability (i.e. p ≥ 0 and p ≤ 1):

${𝒫}=\{p_{\theta }(k)=\frac{n!}{k!(n-k)!}{p}^k(1-p{)}^{n-k},k=0,1,2,\dots ,n|n\in {\mathbb{\mathbb{Z}}}_{\ge 0},p\ge 0\wedge p\le 1\}.$

This example illustrates the definition for a model with some discrete parameters.

General remarks

A parametric model is called identifiable if the mapping θ ↦ P_θ is invertible, i.e. there are no two different parameter values θ₁ and θ₂ such that 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:

• 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. 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.

See also

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