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: $$\{P_\theta: \theta \in \Theta\}$$ 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 $$\Theta \subseteq \mathbb{R}^k$$.
 * 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, $$\Theta \subseteq V$$ 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, $$\Theta \subseteq \mathbb{R}^k \times V$$, 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\left(-\int_0^t \lambda_0(u) e^{\beta x} du\right), $$ where $$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.