Parametric model

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

\mathcal{P} = \big\{ F_\theta\ \big|\ \theta\in\Theta \big\}. $$

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:

\mathcal{P} = \big\{ f_\theta\ \big|\ \theta\in\Theta \big\}. $$

Examples

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

\mathcal{P} = \Big\{\ p_\lambda(j) = \tfrac{\lambda^j}{j!}e^{-\lambda},\ j=0,1,2,3,\dots \ \Big|\;\; \lambda>0 \ \Big\}, $$ 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:

\mathcal{P} = \Big\{\ f_\theta(x) = \tfrac{1}{\sqrt{2\pi}\sigma} \exp\left(-\tfrac{(x-\mu)^2}{2\sigma^2}\right)\ \Big|\;\; \mu\in\mathbb{R}, \sigma>0 \ \Big\}. $$ This parametrized family is both an exponential family and a location-scale family.


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

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


 * 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 $p ≤ 1$):

\mathcal{P} = \Big\{\ p_\theta(k) = \tfrac{n!}{k!(n-k)!}\, p^k (1-p)^{n-k},\ k=0,1,2,\dots, n \ \Big|\;\; n\in\mathbb{Z}_{\ge 0},\, p \ge 0 \land p \le 1\Big\}. $$ 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 $θ_{1}$ and $θ_{2}$ such that $P_{θ_{1}} = P_{θ_{2}}|undefined$.

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.