In statistics, a semiparametric model is a statistical model that has parametric and nonparametric components.
A statistical model is a collection of distributions:
{
P
θ
:
θ
∈
Θ
}
indexed by a parameter
θ
.
A parametric model is one in which the indexing parameter is a finite-dimensional vector (in
k
-dimensional Euclidean space for some integer
k
); i.e. the set of possible values for
θ
is a subset of
R
k
, or
Θ
⊂
R
k
. In this case we say that
θ
is finite-dimensional.
In nonparametric models, the set of possible values of the parameter
θ
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,
Θ
⊂
F
for some possibly infinite-dimensional space
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
Θ
in a semiparametric model satisfies
Θ
⊂
R
k
×
F
, where
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
θ
. 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.
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
(
−
∫
0
t
λ
0
(
u
)
e
β
′
x
d
u
)
,
where
x
is the covariate vector, and
β
and
λ
0
(
u
)
are unknown parameters.
θ
=
(
β
,
λ
0
(
u
)
)
. Here
β
is finite-dimensional and is of interest;
λ
0
(
u
)
is an unknown non-negative function of time (known as the baseline hazard function) and is often a nuisance parameter. The collection of possible candidates for
λ
0
(
u
)
is infinite-dimensional.
