Clock hypothesis

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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:Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

• 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.

Definition

Template:Inline 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

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

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

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

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:

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

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

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

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

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

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

Regular parametric model

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

1. Θ is an open subset of R^k.
2. The map

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

   from Θ to L²(μ) is Fréchet differentiable: there exists a vector $\dot{s}(\theta )=({\dot{s}}_1(\theta ),\dots ,{\dot{s}}_k(\theta ))$ such that

   $\Vert s(\theta +h)-s(\theta )-\dot{s}(\theta {)}^{\prime }h\Vert =o(\left|h\right|)\text{as }h\to 0,$

   where ′ denotes matrix transpose.
3. The map $\theta \mapsto \dot{s}(\theta )$ (defined above) is continuous on Θ.
4. The k×k Fisher information matrix

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

   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:
  1. The density function ƒ_θ(x) is continuously differentiable in θ for μ-almost all x, with gradient ∇ƒ_θ.
  2. The score function

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

     belongs to the space L²(P_θ) of square-integrable functions with respect to the measure P_θ.
  3. The Fisher information matrix I(θ), defined as

     $I_{\theta }=\int z_{\theta }{z_{\theta }}^{\prime }d{P}_{\theta }$

     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 $\sqrt{n}$-consistent and efficient estimator of its parameter θ.

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)

Notes

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References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Template:MR