Semiparametric 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: $\{P_{\theta }:\theta \in \mathrm{\Theta }\}$ 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 $k$. Thus, $\theta$ is finite-dimensional, and $\mathrm{\Theta }\subseteq {\mathbb{ℝ}}^k$.
• 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, $\mathrm{\Theta }\subseteq V$ 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, $\mathrm{\Theta }\subseteq {\mathbb{ℝ}}^k\times V$, 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. 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 $T$ to an event such as death due to cancer or failure of a light bulb, the Cox model specifies the following distribution function for $T$:

$F(t)=1-\exp (-\underset{0}{\overset{t}{\int }}{\lambda }_0(u)e^{\beta x}du),$

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.

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