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: [math]\displaystyle{ \{P_\theta: \theta \in \Theta\} }[/math] indexed by a parameter [math]\displaystyle{ \theta }[/math].

• A parametric model is a model in which the indexing parameter [math]\displaystyle{ \theta }[/math] is a vector in [math]\displaystyle{ k }[/math]-dimensional Euclidean space, for some nonnegative integer [math]\displaystyle{ k }[/math]. Thus, [math]\displaystyle{ \theta }[/math] is finite-dimensional, and [math]\displaystyle{ \Theta \subseteq \mathbb{R}^k }[/math].
• With a nonparametric model, the set of possible values of the parameter [math]\displaystyle{ \theta }[/math] is a subset of some space [math]\displaystyle{ V }[/math], 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, [math]\displaystyle{ \Theta \subseteq V }[/math] for some possibly infinite-dimensional space [math]\displaystyle{ V }[/math].
• 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, [math]\displaystyle{ \Theta \subseteq \mathbb{R}^k \times V }[/math], where [math]\displaystyle{ V }[/math] 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 [math]\displaystyle{ \theta }[/math]. That is, the infinite-dimensional component is regarded as a nuisance parameter. 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 [math]\displaystyle{ T }[/math] to an event such as death due to cancer or failure of a light bulb, the Cox model specifies the following distribution function for [math]\displaystyle{ T }[/math]:

[math]\displaystyle{ F(t) = 1 - \exp\left(-\int_0^t \lambda_0(u) e^{\beta x} du\right), }[/math]

where [math]\displaystyle{ x }[/math] is the covariate vector, and [math]\displaystyle{ \beta }[/math] and [math]\displaystyle{ \lambda_0(u) }[/math] are unknown parameters. [math]\displaystyle{ \theta = (\beta, \lambda_0(u)) }[/math]. Here [math]\displaystyle{ \beta }[/math] is finite-dimensional and is of interest; [math]\displaystyle{ \lambda_0(u) }[/math] 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 [math]\displaystyle{ \lambda_0(u) }[/math] is infinite-dimensional.

See also

• Semiparametric regression
• Statistical model
• Generalized method of moments

References

• Bickel, P. J.; Klaassen, C. A. J.; Ritov, Y.; Wellner, J. A. (1998), Efficient and Adaptive Estimation for Semiparametric Models, Springer 
• Härdle, Wolfgang; Müller, Marlene; Sperlich, Stefan; Werwatz, Axel (2004), Nonparametric and Semiparametric Models, Springer 
• Kosorok, Michael R. (2008), Introduction to Empirical Processes and Semiparametric Inference, Springer 
• Tsiatis, Anastasios A. (2006), Semiparametric Theory and Missing Data, 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