A model is a collection of distributions: indexed by a parameter .
- A parametric model is one in which the indexing parameter is a finite-dimensional vector (in -dimensional Euclidean space for some integer ); i.e. the set of possible values for is a subset of , or . In this case we say that is finite-dimensional.
- In nonparametric models, the set of possible values of the parameter 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, for some possibly infinite-dimensional space .
- 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 in a semiparametric model satisfies , where 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 . 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.
A well-known example of a semiparametric model is the Cox proportional hazards model. 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 collection of possible candidates for is infinite-dimensional.
- N. Balakrishnan; C.R. Rao (30 January 2004). Handbook of Statistics: Advances in Survival Analysis. Elsevier. p. 126. ISBN 978-0-08-049511-8.