Semiparametric model

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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.

  • 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 \theta is a subset of \mathbb{R}^k, or \Theta \subset \mathbb{R}^k. In this case we say that \theta is finite-dimensional.
  • In nonparametric models, the set of possible values of the parameter \theta 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, \Theta \subset \mathbb{F} for some possibly infinite-dimensional space \mathbb{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 \Theta in a semiparametric model satisfies \Theta \subset \mathbb{R}^k \times \mathbb{F}, where \mathbb{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 \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[edit]

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[edit]

References[edit]

  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.