半参数模型
Semiparametric model - Wikipedia https://en.wikipedia.org/wiki/Semiparametric_model
In statistics, a semiparametric model is a statistical model that has parametric and nonparametric components.
A statistical model is a collection of distributions: {\displaystyle \{P_{\theta }:\theta \in \Theta \}}
indexed by a parameter {\displaystyle \theta }
.
- A parametric model is one in which the indexing parameter is a finite-dimensional vector (in {\displaystyle k}
-dimensional Euclidean space for some integer {\displaystyle k}); i.e. the set of possible values for {\displaystyle \theta } is a subset of {\displaystyle \mathbb {R} ^{k}}
, or {\displaystyle \Theta \subset \mathbb {R} ^{k}}
. In this case we say that {\displaystyle \theta } is finite-dimensional.
- In nonparametric models, the set of possible values of the parameter {\displaystyle \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, {\displaystyle \Theta \subset \mathbb {F} }
for some possibly infinite-dimensional space {\displaystyle \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 {\displaystyle \Theta }
in a semiparametric model satisfies {\displaystyle \Theta \subset \mathbb {R} ^{k}\times \mathbb {F} }
, where {\displaystyle \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 {\displaystyle \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.
