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: ${\displaystyle \{P_{\theta }:\theta \in \Theta \}}$ indexed by a parameter ${\displaystyle \theta }$.

• A parametric model is a model in which the indexing parameter ${\displaystyle \theta }$ is a vector in ${\displaystyle k}$-dimensional Euclidean space, for some nonnegative integer ${\displaystyle k}$.^[1] Thus, ${\displaystyle \theta }$ is finite-dimensional, and ${\displaystyle \Theta \subseteq \mathbb {R} ^{k}}$.
• With a nonparametric model, the set of possible values of the parameter ${\displaystyle \theta }$ is a subset of some space ${\displaystyle V}$, 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, ${\displaystyle \Theta \subseteq V}$ for some possibly infinite-dimensional space ${\displaystyle V}$.
• 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, ${\displaystyle \Theta \subseteq \mathbb {R} ^{k}\times V}$, where ${\displaystyle V}$ 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 ${\displaystyle \theta }$. That is, the infinite-dimensional component is regarded as a nuisance parameter.^[2] 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.^[3] If we are interested in studying the time ${\displaystyle T}$ to an event such as death due to cancer or failure of a light bulb, the Cox model specifies the following distribution function for ${\displaystyle T}$:

${\displaystyle F(t)=1-\exp \left(-\int _{0}^{t}\lambda _{0}(u)e^{\beta x}du\right),}$

where ${\displaystyle x}$ is the covariate vector, and ${\displaystyle \beta }$ and ${\displaystyle \lambda _{0}(u)}$ are unknown parameters. ${\displaystyle \theta =(\beta ,\lambda _{0}(u))}$. Here ${\displaystyle \beta }$ is finite-dimensional and is of interest; ${\displaystyle \lambda _{0}(u)}$ 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 ${\displaystyle \lambda _{0}(u)}$ is infinite-dimensional.

See also

• Semiparametric regression
• Statistical model
• Generalized method of moments