Semiparametric model

In statistics, 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 . A parametric model is a model in which the indexing parameter is a vector in -dimensional Euclidean space, for some nonnegative integer . Thus, is finite-dimensional, and . With a nonparametric model, the set of possible values of the parameter is a subset of some space , 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, for some possibly infinite-dimensional space . 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, , 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, the infinite-dimensional component is regarded as a nuisance parameter. 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. 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.