In
statistics
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, a parametric model or parametric family or finite-dimensional model is a particular class of
statistical model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population). A statistical model represents, often in considerably idealized form ...
s. Specifically, a parametric model is a family of
probability distributions that has a finite number of parameters.
Definition
A
statistical model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population). A statistical model represents, often in considerably idealized form ...
is a collection of
probability distributions on some
sample space. We assume that the collection, , is indexed by some set . The set is called the parameter set or, more commonly, the
parameter space The parameter space is the space of possible parameter values that define a particular mathematical model, often a subset of finite-dimensional Euclidean space. Often the parameters are inputs of a function, in which case the technical term for ...
. For each , let denote the corresponding member of the collection; so is a
cumulative distribution function. Then a statistical model can be written as
:
The model is a parametric model if for some positive integer .
When the model consists of absolutely continuous distributions, it is often specified in terms of corresponding
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
s:
:
Examples
* The
Poisson family of distributions is parametrized by a single number :
:
where is the
probability mass function. This family is an
exponential family.
* The
normal family is parametrized by , where is a location parameter and is a scale parameter:
:
This parametrized family is both an
exponential family and a
location-scale family.
* The
Weibull translation model has a three-dimensional parameter :
:
* The
binomial model
In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no q ...
is parametrized by , where is a non-negative integer and is a probability (i.e. and ):
:
This example illustrates the definition for a model with some discrete parameters.
General remarks
A parametric model is called
identifiable if the mapping is invertible, i.e. there are no two different parameter values and such that .
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 parameter
Nuisance (from archaic ''nocence'', through Fr. ''noisance'', ''nuisance'', from Lat. ''nocere'', "to hurt") is a common law tort. It means that which causes offence, annoyance, trouble or injury. A nuisance can be either public (also "commo ...
s;
* 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.
See also
*
Parametric family
In mathematics and its applications, a parametric family or a parameterized family is a family of objects (a set of related objects) whose differences depend only on the chosen values for a set of parameters.
Common examples are parametrized (fam ...
*
Parametric statistics
Parametric statistics is a branch of statistics which assumes that sample data comes from a population that can be adequately modeled by a probability distribution that has a fixed set of parameters. Conversely a non-parametric model does not as ...
*
Statistical model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population). A statistical model represents, often in considerably idealized form ...
*
Statistical model specification
In statistics, model specification is part of the process of building a statistical model: specification consists of selecting an appropriate functional form for the model and choosing which variables to include. For example, given personal incom ...
Notes
Bibliography
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{{DEFAULTSORT:Parametric Model
Parametric statistics
Statistical models