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 semiparametric model is 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 ...
that has
parametric and
nonparametric components.
A statistical model is a
parameterized family of distributions:
indexed by a
parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
.
* A
parametric model
In statistics, a parametric model or parametric family or finite-dimensional model is a particular class of statistical models. Specifically, a parametric model is a family of probability distributions that has a finite number of parameters.
Def ...
is a model in which the indexing parameter
is a vector in
-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
, for some nonnegative integer
.
[.] Thus,
is finite-dimensional, and
.
* With a
nonparametric model, the set of possible values of the parameter
is a subset of some space
, which is not necessarily finite-dimensional. For example, we might consider the set of all distributions with mean 0. Such spaces are
vector spaces with topological structure, but may not be finite-dimensional as vector spaces. Thus,
for some possibly
infinite-dimensional space .
* With a semiparametric model, the parameter has both a finite-dimensional component and an infinite-dimensional component (often a real-valued function defined on the real line). Thus,
, where
is an infinite-dimensional space.
It may appear at first that semiparametric models include nonparametric models, since they have an infinite-dimensional as well as a finite-dimensional component. However, a semiparametric model is considered to be "smaller" than a completely nonparametric model because we are often interested only in the finite-dimensional component of
. That is, the infinite-dimensional component is regarded as a
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 ...
.
[.] In nonparametric models, by contrast, the primary interest is in estimating the infinite-dimensional parameter. Thus the estimation task is statistically harder in nonparametric models.
These models often use
smoothing or
kernels.
Example
A well-known example of a semiparametric model is the
Cox proportional hazards model.
If we are interested in studying the time
to an event such as death due to cancer or failure of a light bulb, the Cox model specifies the following distribution function for
:
:
where
is the covariate vector, and
and
are unknown parameters.
. Here
is finite-dimensional and is of interest;
is an unknown non-negative function of time (known as the baseline hazard function) and is often a
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 ...
. The set of possible candidates for
is infinite-dimensional.
See also
*
Semiparametric regression
*
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 ...
*
Generalized method of moments
Notes
References
*
*
*
*{{citation , first1= Anastasios A. , last1= Tsiatis , title= Semiparametric Theory and Missing Data , year= 2006 , publisher= Springer
*Begun, Janet M.; Hall, W. J.; Huang, Wei-Min; Wellner, Jon A. (1983), "Information and asymptotic efficiency in parametric--nonparametric models", Annals of Statistics, 11 (1983), no. 2, 432--452
Mathematical and quantitative methods (economics)