Instatistics, a semiparametric model is a statistical model that has parametric and nonparametric components.
A statistical model is a parameterized family of distributions: indexed by a parameter
.
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 . 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 smoothingorkernels.
A well-known example of a semiparametric model is the Cox proportional hazards model.[3] If we are interested in studying the time to an event such as death due to cancer or failure of a light bulb, the Cox model specifies the following distribution function for
:
where is the covariate vector, and
and
are unknown parameters.
. Here
is finite-dimensional and is of interest;
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
is infinite-dimensional.