Parametric Statistical Model Family

A Parametric Statistical Model Family is a statistical model family that is a parametric model (which assumes a distribution in the sample space described by a low-order vector).

• AKA: Parametric Probability Distribution.
• Context:
  • It can range from being a One-Parameter Statistical Model Family to being a Two-Parameter Statistical Model Family to being a Multi-Paramater Statistical Model Family.
  • It can range from being a Fully-Parametric Statistical Model to being a Semi-Parametric Statistical Model.
  • It can be trained by a Parametric Model Training Task.
• Example(s):
  • a Linear Regression Model.
  • a Poisson Distribution.
  • a Gaussian Distribution.
  • …
• Counter-Example(s):
  • Nonparametric Model.
  • a Nonparametric Statistical Model/Non-Parametric Probability Distribution.
  • a Semi-Nonparametric Model.
• See: Probability Distribution Parameter Estimation, Generative Learning, Indexed Family, Generative Model, Parametric Regression Algorithm.

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References

2014

• (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/Parametric_family Retrieved:2014-12-8.
  • In mathematics and its applications, a parametric family or a parameterized family is a family of objects (a set of related objects) whose definitions depend on a set of parameters.

    Common examples are parametrized (families of) functions, probability distributions, curves, shapes, etc.

2013

• http://en.wikipedia.org/wiki/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:^{[citation needed]}

    • in a “parametric” model all the parameters are in finite-dimensional parameter spaces;
    • a model is “nonparametric” 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.

• http://en.wikipedia.org/wiki/Parametric_model#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 ⊆ Rk, and the model itself is written as : [math]\displaystyle{ \mathcal{P} = \big\{ P_\theta\ \big|\ \theta\in\Theta \big\}. }[/math] When the model consists of absolutely continuous distributions, it is often specified in terms of corresponding probability density functions: : [math]\displaystyle{ \mathcal{P} = \big\{ f_\theta\ \big|\ \theta\in\Theta \big\}. }[/math] 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.

2012

• http://reference.wolfram.com/language/guide/ParametricStatisticalDistributions.html
  • QUOTE: In almost every area where probability and statistics are used there have been found a few parametric distribution families that are known to be good models. The origins vary from combinatorial arguments, such as in urn models, to transformations of existing distributions, or as different kinds of limit processes. The collection of parametric distributions in the Wolfram Language has been selected in order to provide complete modeling frameworks for a variety of areas. The result is the most extensive collection of parametric distributions ever assembled. From a distribution there are dozens of properties, such as distribution functions, moments, or quantiles, that are directly accessible. Parametric distributions are used as arguments to higher-level functions that compute probabilities, expectations, random variates, or parameter estimates from data. Distributions with undetermined parameters can be used throughout, and later the parameters can be solved for or optimized over, etc.

2008

• (Georgii, 2008) ⇒ Hans-Otto Georgii. (2008). “Stochastics: introduction to probability theory and statistics." Walter de Gruyter. ISBN:3110191458,
  • QUOTE: A statistical model [math]\displaystyle{ M = (X, F, P_v : v \in \theta) }[/math] is called a parametric model if [math]\displaystyle{ \theta \in \mathbb{R}^d }[/math] for some [math]\displaystyle{ d \in \N }[/math]. For [math]\displaystyle{ d=1 }[/math], [math]\displaystyle{ M }[/math] is called a one-parameter model.

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