Semiparametric model

In statistics a semiparametric model is a model that has parametric and nonparametric components.

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


 * 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 $$\theta$$ is a subset of $$\mathbb{R}^k$$, or $$\Theta \subset \mathbb{R}^k$$. In this case we say that $$\theta$$ is finite-dimensional.
 * In nonparametric models, the set of possible values of the parameter $$\theta$$ 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, $$\Theta \subset \mathbb{F}$$ for some possibly infinite dimensional space $$\mathbb{F}$$.
 * 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 $$\Theta$$ in a semiparametric model satisfies $$\Theta \subset \mathbb{R}^k \times \mathbb{F}$$, where $$\mathbb{F}$$ 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 $$\theta$$. 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.

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\left(-\int_0^t \lambda_0(u) e^{\beta'x(u)} du\right), $$ where $$x(u)$$ is a known function of time (the covariate vector at time $$u$$), and $$\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)$$ is an unknown non-negative function of time (known as the baseline hazard function) and is often a nuisance parameter. The collection of possible candidates for $$\lambda_0(u)$$ is infinite dimensional.