In statistics, a semiparametric model is a statistical model that has parametric and nonparametric components.
A statistical model is a collection of distributions: { P θ : θ ∈ Θ } indexed by a parameter θ .
A parametric model is one in which the indexing parameter is a finite-dimensional vector (in k -dimensional Euclidean space for some integer k ); i.e. the set of possible values for θ is a subset of R k , or Θ ⊂ R k . In this case we say that θ is finite-dimensional.In nonparametric models, the set of possible values of the parameter θ is a subset of some space, 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 ⊂ F for some possibly infinite-dimensional space F .In semiparametric models, the parameter has both a finite-dimensional component and an infinite-dimensional component (often a real-valued function defined on the real line). Thus the parameter space Θ in a semiparametric model satisfies Θ ⊂ R k × F , where F 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 . That is, we are not interested in estimating the infinite-dimensional component. 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.
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 ( − ∫ 0 t λ 0 ( u ) e β ′ x d u ) , where x is the covariate vector, and β and λ 0 ( u ) are unknown parameters. θ = ( β , λ 0 ( u ) ) . Here β is finite-dimensional and is of interest; λ 0 ( u ) is an unknown non-negative function of time (known as the baseline hazard function) and is often a nuisance parameter. The collection of possible candidates for λ 0 ( u ) is infinite-dimensional.
