Semiparametric model

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

A model is a collection of distributions: \{P_\theta: \theta \in \Theta\} indexed by a parameter \theta.

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, 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.

Example

A well-known example of a semiparametric model is the Cox proportional hazards model.[1] 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_0^t \lambda_0(u) e^{\beta'x} du\right),

where x is the covariate vector, and \beta and \lambda_0(u) are unknown parameters. \theta = (\beta, \lambda_0(u)). Here \beta is finite-dimensional and is of interest; \lambda_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 \lambda_0(u) is infinite-dimensional.

See also

References

  1. N. Balakrishnan; C.R. Rao (30 January 2004). Handbook of Statistics: Advances in Survival Analysis. Elsevier. p. 126. ISBN 978-0-08-049511-8.
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