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: ${\displaystyle \{P_{\theta }:\theta \in \Theta \}}${\displaystyle \{P_{\theta }:\theta \in \Theta \}} indexed by a parameter ${\displaystyle \theta }${\displaystyle \theta }.

• A parametric model is a model in which the indexing parameter ${\displaystyle \theta }${\displaystyle \theta } is a vector in ${\displaystyle k}${\displaystyle k}-dimensional Euclidean space, for some nonnegative integer ${\displaystyle k}${\displaystyle k}.^[1] Thus, ${\displaystyle \theta }${\displaystyle \theta } is finite-dimensional, and ${\displaystyle \Theta \subseteq \mathbb {R} ^{k}}${\displaystyle \Theta \subseteq \mathbb {R} ^{k}}.
• With a nonparametric model, the set of possible values of the parameter ${\displaystyle \theta }${\displaystyle \theta } is a subset of some space ${\displaystyle V}${\displaystyle 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, ${\displaystyle \Theta \subseteq V}${\displaystyle \Theta \subseteq V} for some possibly infinite-dimensional space ${\displaystyle V}${\displaystyle 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, ${\displaystyle \Theta \subseteq \mathbb {R} ^{k}\times V}${\displaystyle \Theta \subseteq \mathbb {R} ^{k}\times V}, where ${\displaystyle V}${\displaystyle 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 ${\displaystyle \theta }${\displaystyle \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.

A well-known example of a semiparametric model is the Cox proportional hazards model.^[3] If we are interested in studying the time ${\displaystyle T}${\displaystyle 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 ${\displaystyle T}${\displaystyle T}:

${\displaystyle F(t)=1-\exp \left(-\int _{0}^{t}\lambda _{0}(u)e^{\beta x}du\right),}${\displaystyle F(t)=1-\exp \left(-\int _{0}^{t}\lambda _{0}(u)e^{\beta x}du\right),}

where ${\displaystyle x}${\displaystyle x} is the covariate vector, and ${\displaystyle \beta }${\displaystyle \beta } and ${\displaystyle \lambda _{0}(u)}${\displaystyle \lambda _{0}(u)} are unknown parameters. ${\displaystyle \theta =(\beta ,\lambda _{0}(u))}${\displaystyle \theta =(\beta ,\lambda _{0}(u))}. Here ${\displaystyle \beta }${\displaystyle \beta } is finite-dimensional and is of interest; ${\displaystyle \lambda _{0}(u)}${\displaystyle \lambda _{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 ${\displaystyle \lambda _{0}(u)}${\displaystyle \lambda _{0}(u)} is infinite-dimensional.

• Semiparametric regression
• Statistical model
• Generalized method of moments

• Bickel, P. J.; Klaassen, C. A. J.; Ritov, Y.; Wellner, J. A. (1998), Efficient and Adaptive Estimation for Semiparametric Models, Springer
• Härdle, Wolfgang; Müller, Marlene; Sperlich, Stefan; Werwatz, Axel (2004), Nonparametric and Semiparametric Models, Springer
• Kosorok, Michael R. (2008), Introduction to Empirical Processes and Semiparametric Inference, Springer
• Tsiatis, Anastasios A. (2006), Semiparametric Theory and Missing Data, Springer
• 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

• Semi-parametric models
• Mathematical and quantitative methods (economics)

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