Semiparametric model explained

In statistics, a semiparametric model is a statistical model that has parametric and nonparametric components.

A statistical model is a parameterized family of distributions:

\{P _{\theta:} \theta \in \Theta\} indexed by a parameter \theta .

• A parametric model is a model in which the indexing parameter

\theta is a vector in k -dimensional Euclidean space, for some nonnegative integer k .^[1] Thus, \theta is finite-dimensional, and \Theta \subseteq R ^{k} .

• With a nonparametric model, the set of possible values of the parameter

\theta is a subset of some space 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, \Theta \subseteq V for some possibly infinite-dimensional space 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,

\Theta \subseteq R ^{k} x V , where 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

\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

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\left( - \int ^{t} _{0} λ _{0(u)} e ^{\beta} du\right), where x is the covariate vector, and \beta and λ _{0(u)} are unknown parameters. \theta = (\beta, λ _{0(u))} . Here \beta 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 set of possible candidates for λ _{0(u)} is infinite-dimensional.

See also

• Semiparametric regression
• Statistical model
• Generalized method of moments

References

• • • • • Begun, Janet M.; Hall, W. J.; Huang, Wei-Min; Wellner, Jon A. (1983), "Information and asymptotic efficiency in parametric--nonparametric models", Annals of Statistics, 11 (1983), no. 2, 432--452

Notes and References

1. .
2. .
3. Book: N. . Balakrishnan. C. R. . Rao . C. R. Rao. Handbook of Statistics 23: Advances in Survival Analysis. 2004 . Elsevier. 126.