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In
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, 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 (statistics), sample data (and similar data from a larger Statistical population, population). A statistical model repres ...
that has parametric and nonparametric components. A statistical model is a
parameterized family In mathematics and its applications, a parametric family or a parameterized family is a indexed family, family of objects (a set of related objects) whose differences depend only on the chosen values for a set of parameters. Common examples are p ...
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 ...
\theta. * 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 \theta is a vector in k-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, for some nonnegative integer k.. Thus, \theta is finite-dimensional, and \Theta \subseteq \mathbb^k. * With a nonparametric model, the set of possible values of the parameter \theta is a subset of some space V, 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, \Theta \subseteq V for some possibly infinite-dimensional space V. * 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, \Theta \subseteq \mathbb^k \times V, where V 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 \theta. 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 "common") ...
.. 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 Proportional hazards models are a class of survival models in statistics. Survival models relate the time that passes, before some event occurs, to one or more covariates that may be associated with that quantity of time. In a proportional hazar ...
. If we are interested in studying the time T to an event such as death due to cancer or failure of a light bulb, the Cox model specifies the following distribution function for T: : F(t) = 1 - \exp\left(-\int_0^t \lambda_0(u) e^ du\right), where x is the covariate vector, and \beta and \lambda_0(u) are unknown parameters. \theta = (\beta, \lambda_0(u)). Here \beta is finite-dimensional and is of interest; \lambda_0(u) 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 "common") ...
. The set of possible candidates for \lambda_0(u) 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 (statistics), sample data (and similar data from a larger Statistical population, population). A statistical model repres ...
* 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)