Semiparametric model
# 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.

