Semiparametric model

In <a href="/facts/Statistics/DNPsKGYU">statistics</a>, a semiparametric model is a <a href="/facts/Statistical_model/jKkT5ftm">statistical model</a> that has <a href="/facts/Parametric_statistics/Uvq9SJV9">parametric</a> and <a href="/facts/Nonparametric/svTDJHTb">nonparametric</a> components.
A statistical model is a <a href="/facts/Parameterized_family/YoEWw0I8">parameterized family</a> of distributions: 
 
 
 
 {
 
 P
 
 θ
 
 
 :
 θ
 ∈
 Θ
 }
 
 
 {\displaystyle \{P_{\theta }:\theta \in \Theta \}}
 
 indexed by a <a href="/facts/Statistical_parameter/fkJju25M">parameter</a> 
 
 
 
 θ
 
 
 {\displaystyle \theta }
 
. 

<ul><li>A <a href="/facts/Parametric_model/FeynAa0f">parametric model</a> is a model in which the indexing parameter 
 
 
 
 θ
 
 
 {\displaystyle \theta }
 
 is a vector in 
 
 
 
 k
 
 
 {\displaystyle k}
 
-dimensional <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a>, for some nonnegative integer 
 
 
 
 k
 
 
 {\displaystyle k}
 
. Thus, 
 
 
 
 θ
 
 
 {\displaystyle \theta }
 
 is finite-dimensional, and 
 
 
 
 Θ
 ⊆
 
 
 R
 
 
 k
 
 
 
 
 {\displaystyle \Theta \subseteq \mathbb {R} ^{k}}
 
.</li>
<li>With a <a href="/facts/Non-parametric_statistics/svTDJHTb">nonparametric model</a>, the set of possible values of the parameter 
 
 
 
 θ
 
 
 {\displaystyle \theta }
 
 is a subset of some space 
 
 
 
 V
 
 
 {\displaystyle V}
 
, which is not necessarily finite-dimensional. For example, we might consider the set of all distributions with mean 0. Such spaces are <a href="/facts/Topological_vector_space/XRex1ChN">vector spaces with topological structure</a>, but may not be finite-dimensional as vector spaces. Thus, 
 
 
 
 Θ
 ⊆
 V
 
 
 {\displaystyle \Theta \subseteq V}
 
 for some possibly <a href="/facts/Infinite-dimensional_vector_space/z8m02A3W">infinite-dimensional space</a> 
 
 
 
 V
 
 
 {\displaystyle V}
 
.</li>
<li>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, 
 
 
 
 Θ
 ⊆
 
 
 R
 
 
 k
 
 
 ×
 V
 
 
 {\displaystyle \Theta \subseteq \mathbb {R} ^{k}\times V}
 
, where 
 
 
 
 V
 
 
 {\displaystyle V}
 
 is an infinite-dimensional space.</li></ul>
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 
 
 
 
 θ
 
 
 {\displaystyle \theta }
 
. That is, the infinite-dimensional component is regarded as a <a href="/facts/Nuisance_parameter/acJWVqxj">nuisance parameter</a>. 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 <a href="/facts/Smoothing/2svFjsbI">smoothing</a> or <a href="/facts/Kernel_(statistics)/otWLlBbA">kernels</a>.

Semiparametric model open-in-new

Semiparametric model