In
statisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, a
parametric model or
parametric family or
finitedimensional model is a family of
distributionIn probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
s that can be described using a finite number of
parameterParameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....
s. These parameters are usually collected together to form a single
kdimensional
parameter vector θ = (
θ_{1},
θ_{2}, …,
θ_{k}).
Parametric models are contrasted with the
semiparametric, seminonparametric, and nonparametric 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 finitedimensional parameter spaces;
 a model is “nonparametric” if all the parameters are in infinitedimensional parameter spaces;
 a “semiparametric” model contains finitedimensional parameters of interest and infinitedimensional nuisance parameters;
 a “seminonparametric” model has both finitedimensional and infinitedimensional unknown parameters of interest.
Some statisticians believe that the concepts “parametric”, “nonparametric”, and “semiparametric” 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
A
parametric model is a collection of
probability distributionIn probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
s such that each member of this collection,
P_{θ}, is described by a finitedimensional parameter
θ. The set of all allowable values for the parameter is denoted Θ ⊆
R^{k}, and the model itself is written as

When the model consists of absolutely continuous distributions, it is often specified in terms of corresponding probability density functionIn probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
s:

The parametric model is called identifiable if the mapping θ ↦ P_{θ} is invertible, that is there are no two different parameter values θ_{1} and θ_{2} such that P_{θ1} = P_{θ2}.
Examples
 The Poisson family
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since...
of distributions is parametrized by a single number λ > 0:

where p_{λ} is the probability mass functionIn probability theory and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value...
. This family is an exponential familyIn probability and statistics, an exponential family is an important class of probability distributions sharing a certain form, specified below. This special form is chosen for mathematical convenience, on account of some useful algebraic properties, as well as for generality, as exponential...
.
 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
In probability and statistics, an exponential family is an important class of probability distributions sharing a certain form, specified below. This special form is chosen for mathematical convenience, on account of some useful algebraic properties, as well as for generality, as exponential...
and a locationscale familyIn probability theory, especially as that field is used in statistics, a locationscale family is a family of univariate probability distributions parametrized by a location parameter and a nonnegative scale parameter; if X is any random variable whose probability distribution belongs to such a...
:

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

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 σfinite measure on a probability spaceIn probability theory, a probability space or a probability triple is a mathematical construct that models a realworld process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...
(Ω, ℱ), and the collection of all probability measures dominated by μ. Then we will call $\backslash mathcal\{P\}\backslash !=\backslash !\backslash \{\; P\_\backslash theta\backslash ,\; \backslash theta\backslash in\backslash Theta\; \backslash \}\; \backslash subseteq\; \backslash mathcal\{M\}\_\backslash mu$ a regular parametric model if the following requirements are met:
 Θ is an open subset
The concept of an open set is fundamental to many areas of mathematics, especially pointset topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
of R^{k}.
 The map

from Θ to L^{2}(μ) is Fréchet differentiableIn mathematics, the Fréchet derivative is a derivative defined on Banach spaces. Named after Maurice Fréchet, it is commonly used to formalize the concept of the functional derivative used widely in the calculus of variations. Intuitively, it generalizes the idea of linear approximation from...
: there exists a vector $\backslash dot\{s\}(\backslash theta)\; =\; (\backslash dot\{s\}\_1(\backslash theta),\backslash ,\backslash ldots,\backslash ,\backslash dot\{s\}\_k(\backslash theta))$ such that

where ′ denotes matrix transposeIn linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...
.
 The map $\backslash theta\backslash mapsto\backslash dot\{s\}(\backslash theta)$ (defined above) is continuous
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
on Θ.
 The k×k Fisher information matrix

is nonsingular.
Properties
See also
 Statistical model
A statistical model is a formalization of relationships between variables in the form of mathematical equations. A statistical model describes how one or more random variables are related to one or more random variables. The model is statistical as the variables are not deterministically but...
 Parametric family
In mathematics and its applications, a parametric family or a parameterized family is a family of objects whose definitions depend on a set of parameters....
 Parametrization
Parametrization is the process of deciding and defining the parameters necessary for a complete or relevant specification of a model or geometric object....
(i.e., coordinate systemIn geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...
)
 Parsimony (with regards to the tradeoff of many or few parameters in data fitting)