Parametric model

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, …, θ_k).

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.^[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, P_θ, is described by a finite-dimensional parameter θ. The set of all allowable values for the parameter is denoted Θ ⊆ R^k, and the model itself is written as

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

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

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

The parametric model is called identifiable if the mapping θ ↦ P_θ is invertible, that is there are no two different parameter values θ₁ and θ₂ such that P_θ1 = P_θ2.

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

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

• The Weibull translation model has three parameters θ = (λ, β, μ):

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

  This model is not regular (see definition below) unless we restrict β to lie in the interval (2, +∞).

Regular parametric model

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

1. $\Theta$ is an open subset of ${\displaystyle \mathbb {R} ^{k}}$.
2. The map

   $\theta\mapsto s(\theta)=\sqrt{dP_\theta/d\mu}$

   from $\Theta$ to ${\displaystyle L^{2}(\mu )}$ is Fréchet differentiable: there exists a vector $\dot{s}(\theta) = (\dot{s}_1(\theta),\,\ldots,\,\dot{s}_k(\theta))$ 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 $\theta\mapsto\dot{s}(\theta)$ (defined above) is continuous on $\Theta$.
4. The $k\times k$ Fisher information matrix

   $I(\theta) = 4\int \dot{s}(\theta)\dot{s}(\theta)'d\mu$

   is non-singular.

Properties

• Sufficient conditions for regularity of a parametric model in terms of ordinary differentiability of the density function ƒ_θ are following:^[4]
  1. The density function ƒ_θ(x) is continuously differentiable in θ for μ-almost all $x$, with gradient ${\displaystyle \nabla f_{\theta }}$.
  2. The score function

     $z_\theta = \frac{\nabla f_\theta}{f_\theta} \cdot \mathbf{1}_{\{f_\theta>0\}}$

     belongs to the space ${\displaystyle L^{2}(P_{\theta })}$ of square-integrable functions with respect to the measure $P_{\theta }$.

  3. The Fisher information matrix I(θ), defined as

     $I_\theta = \int \!z_\theta z_\theta' \,dP_\theta$

     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 $\scriptstyle \sqrt{n}$-consistent and efficient estimator of its parameter θ.^[5]

See also

• Statistical model
• Parametric family
• Parametrization (i.e., coordinate system)
• Parsimony (with regards to the trade-off of many or few parameters in data fitting)
• Parametricism

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