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 θ : θ ∈ Θ } {\displaystyle \{P_{\theta }:\theta \in \Theta \}} indexed by a parameter θ {\displaystyle \theta } .
A parametric model is a model in which the indexing parameter θ {\displaystyle \theta } is a vector in k {\displaystyle k} -dimensional Euclidean space, for some nonnegative integer k {\displaystyle k} . Thus, θ {\displaystyle \theta } is finite-dimensional, and Θ ⊆ R k {\displaystyle \Theta \subseteq \mathbb {R} ^{k}} . With a nonparametric model, the set of possible values of the parameter θ {\displaystyle \theta } is a subset of some space V {\displaystyle 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, Θ ⊆ V {\displaystyle \Theta \subseteq V} for some possibly infinite-dimensional space V {\displaystyle 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, Θ ⊆ R k × V {\displaystyle \Theta \subseteq \mathbb {R} ^{k}\times V} , where V {\displaystyle 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 θ {\displaystyle \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 {\displaystyle 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 {\displaystyle T} :
F ( t ) = 1 − exp ( − ∫ 0 t λ 0 ( u ) e β x d u ) , {\displaystyle F(t)=1-\exp \left(-\int _{0}^{t}\lambda _{0}(u)e^{\beta x}du\right),}
where x {\displaystyle x} is the covariate vector, and β {\displaystyle \beta } and λ 0 ( u ) {\displaystyle \lambda _{0}(u)} are unknown parameters. θ = ( β , λ 0 ( u ) ) {\displaystyle \theta =(\beta ,\lambda _{0}(u))} . Here β {\displaystyle \beta } is finite-dimensional and is of interest; λ 0 ( u ) {\displaystyle \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 λ 0 ( u ) {\displaystyle \lambda _{0}(u)} is infinite-dimensional.
See also Semiparametric regression Statistical model Generalized method of moments
Notes
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
