semiparametric model

{{Short description|Type of statistical model}}In statistics, a semiparametric model is a statistical model that has parametric and nonparametric components.

A statistical model is a parameterized family of distributions: $\{P_\theta: \theta \in \Theta\}$ indexed by a parameter $\theta$ .

A parametric model is a model in which the indexing parameter $\theta$ is a vector in $k$ -dimensional Euclidean space, for some nonnegative integer $k$ .{{citation| title= Semiparametrics | last1= Bickel | first1= P. J. | last2= Klaassen | first2= C. A. J. | last3= Ritov | first3= Y. | last4= Wellner | first4= J. A. | encyclopedia= Encyclopedia of Statistical Sciences | editor1-first= S. | editor1-last= Kotz | editor1-link= Samuel Kotz |display-editors=etal | year= 2006 | publisher= Wiley}}. Thus, $\theta$ is finite-dimensional, and $\Theta \subseteq \mathbb{R}^k$ .
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, $\Theta \subseteq \mathbb{R}^k \times V$ , 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.{{citation| title= Semi-parametric models | first= D. | last= Oakes | encyclopedia= Encyclopedia of Statistical Sciences | editor1-first= S. | editor1-last= Kotz | editor1-link= Samuel Kotz |display-editors=etal | year= 2006 | publisher= Wiley}}. 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.{{cite book|author1-first=N. | author1-last= Balakrishnan|author2-first=C. R. | author2-last=Rao | author2-link= C. R. Rao|title=Handbook of Statistics 23: Advances in Survival Analysis|url=https://books.google.com/books?id=oP4ZJxBE1csC&pg=PA126|date= 2004 |publisher= Elsevier|pages=126}} 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 $T$ :

: $F(t) = 1 - \exp\left(-\int_0^t \lambda_0(u) e^{\beta x} du\right),$

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. The set of possible candidates for $\lambda_0(u)$ is infinite-dimensional.

Notes

References

{{citation | last1= Bickel | first1= P. J. | last2= Klaassen | first2= C. A. J. | last3= Ritov | first3= Y. | last4= Wellner | first4= J. A. | year= 1998 | title= Efficient and Adaptive Estimation for Semiparametric Models | publisher= Springer}}
{{citation | first1= Wolfgang | last1= Härdle | first2= Marlene | last2= Müller |first3= Stefan | last3= Sperlich | first4=Axel | last4= Werwatz | title= Nonparametric and Semiparametric Models | year= 2004 | publisher= Springer}}
{{citation | first1= Michael R. | last1= Kosorok | title= Introduction to Empirical Processes and Semiparametric Inference | year= 2008 | publisher= Springer}}
{{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

Category:Mathematical and quantitative methods (economics)

semiparametric model

Example

See also

Notes

References