# Parametric model

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This article is about statistics. For mathematical and computer representation of objects, see Solid modeling.

In statistics, a parametric model or parametric family or finite-dimensional model is a family of distributions that can be described using a finite number of parameters. These parameters are usually collected together to form a single k-dimensional parameter vector θ = (θ1, θ2, …, θk).

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]

• 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.

## Definition

A parametric model is a collection of probability distributions such that each member of this collection, Pθ, is described by a finite-dimensional parameter θ. The set of all allowable values for the parameter is denoted Θ ⊆ Rk, and the model itself is written as

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

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

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

The parametric model is called identifiable if the mapping θPθ is invertible, that is there are no two different parameter values θ1 and θ2 such that Pθ1 = Pθ2.

### Examples

• The Poisson family of distributions is parametrized by a single number λ > 0:
${\displaystyle {\mathcal {P}}={\Big \{}\ p_{\lambda }(j)={\tfrac {\lambda ^{j}}{j!}}e^{-\lambda },\ j=0,1,2,3,\dots \ {\Big |}\ \lambda >0\ {\Big \}},}$
where pλ is the probability mass function. This family is an exponential family.
• The normal family is parametrized by θ = (μ,σ), where μR is a location parameter, and σ > 0 is a scale parameter. This parametrized family is both an exponential family and a location-scale family:
${\displaystyle {\mathcal {P}}={\Big \{}\ f_{\theta }(x)={\tfrac {1}{{\sqrt {2\pi }}\sigma }}e^{-{\frac {1}{2\sigma ^{2}}}(x-\mu )^{2}}\ {\Big |}\ \mu \in \mathbb {R} ,\sigma >0\ {\Big \}}.}$
• The Weibull translation model has three parameters θ = (λ, β, μ):
${\displaystyle {\mathcal {P}}={\Big \{}\ f_{\theta }(x)={\tfrac {\beta }{\lambda }}\left({\tfrac {x-\mu }{\lambda }}\right)^{\beta -1}\!\exp \!{\big (}\!-\!{\big (}{\tfrac {x-\mu }{\lambda }}{\big )}^{\beta }{\big )}\,\mathbf {1} _{\{x>\mu \}}\ {\Big |}\ \lambda >0,\,\beta >0,\,\mu \in \mathbb {R} \ {\Big \}}.}$
This model is not regular (see definition below) unless we restrict β to lie in the interval (2, +∞).

## Regular parametric model

Let ${\displaystyle \mu }$ be a fixed σ-finite measure on a measurable space ${\displaystyle (\Omega ,{\mathcal {F}})}$, and ${\displaystyle \scriptstyle {\mathcal {M}}_{\mu }}$ the collection of all probability measures dominated by ${\displaystyle \mu }$. Then we will call ${\displaystyle {\mathcal {P}}\!=\!\{P_{\theta }|\,\theta \in \Theta \}\subseteq {\mathcal {M}}_{\mu }}$ a regular parametric model if the following requirements are met:[3]

1. ${\displaystyle \Theta }$ is an open subset of ${\displaystyle \mathbb {R} ^{k}}$.
2. The map
${\displaystyle \theta \mapsto s(\theta )={\sqrt {dP_{\theta }/d\mu }}}$
from ${\displaystyle \Theta }$ to ${\displaystyle L^{2}(\mu )}$ is Fréchet differentiable: there exists a vector ${\displaystyle {\dot {s}}(\theta )=({\dot {s}}_{1}(\theta ),\,\ldots ,\,{\dot {s}}_{k}(\theta ))}$ such that
${\displaystyle \lVert s(\theta +h)-s(\theta )-{\dot {s}}(\theta )'h\rVert =o(|h|){\text{ as }}h\to 0,}$
where ′ denotes matrix transpose.
3. The map ${\displaystyle \theta \mapsto {\dot {s}}(\theta )}$ (defined above) is continuous on ${\displaystyle \Theta }$.
4. The ${\displaystyle k\times k}$ Fisher information matrix
${\displaystyle I(\theta )=4\int {\dot {s}}(\theta ){\dot {s}}(\theta )'d\mu }$
is non-singular.

### Properties

• Sufficient conditions for regularity of a parametric model in terms of ordinary differentiability of the density function ƒθ are following:[4]
1. The density function ƒθ(x) is continuously differentiable in θ for μ-almost all ${\displaystyle x}$, with gradient ${\displaystyle \nabla f_{\theta }}$.
2. The score function
${\displaystyle z_{\theta }={\frac {\nabla f_{\theta }}{f_{\theta }}}\cdot \mathbf {1} _{\{f_{\theta }>0\}}}$
belongs to the space ${\displaystyle L^{2}(P_{\theta })}$ of square-integrable functions with respect to the measure ${\displaystyle P_{\theta }}$.
3. The Fisher information matrix I(θ), defined as
${\displaystyle I_{\theta }=\int \!z_{\theta }z_{\theta }'\,dP_{\theta }}$
is nonsingular and continuous in θ.

If conditions (i)−(iii) hold then the parametric model is regular.

• Local asymptotic normality.
• If the regular parametric model is identifiable then there exists a uniformly ${\displaystyle \scriptstyle {\sqrt {n}}}$-consistent and efficient estimator of its parameter θ.[5]

## Notes

1. ^ LeCam 2000, ch.7.4
2. ^ Bickel 1998, p. 2
3. ^ Bickel 1998, p. 12
4. ^ Bickel 1998, p.13, prop.2.1.1
5. ^ Bickel 1998, Theorems 2.5.1, 2.5.2

## Bibliography

• Bickel, Peter J. & Doksum, Kjell A. (2001). Mathematical Statistics: Basic and Selected Topics. Volume 1 (Second (updated printing 2007) ed.). Pearson Prentice-Hall.
• Bickel, Peter J.; Klaassen, Chris A.J.; Ritov, Ya’acov; Wellner Jon A. (1998). Efficient and adaptive estimation for semiparametric models. Springer: New York. ISBN 0-387-98473-9.
• Davidson, A.C. (2003). Statistical Models. Cambridge University Press.
• Freedman, David A. (2009). Statistical Models: Theory and Practice (Second ed.). Cambridge University Press. ISBN 978-0-521-67105-7.
• Le Cam, Lucien; Lo Yang, Grace (2000). Asymptotics in statistics: some basic concepts. Springer. ISBN 0-387-95036-2.
• Lehmann, Erich (1983). Theory of Point Estimation.
• Lehmann, Erich (1959). Testing Statistical Hypotheses.
• Liese, Friedrich & Miescke, Klaus-J. (2008). Statistical Decision Theory: Estimation, Testing, and Selection. Springer.
• Pfanzagl, Johann; with the assistance of R. Hamböker (1994). Parametric Statistical Theory. Walter de Gruyter. ISBN 3-11-013863-8. MR 1291393