Semiparametric model

In statistics, a semiparametric model is a statistical model that has parametric and nonparametric components. In statistics, a semiparametric model is a statistical model that has parametric and nonparametric components. A statistical model is a parameterized family of distributions: { P θ : θ ∈ Θ } {displaystyle {P_{ heta }: heta in Theta }} indexed by a parameter θ {displaystyle heta } . 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 θ {displaystyle heta } . 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 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 T {displaystyle T} : where x {displaystyle x} is the covariate vector, and β {displaystyle eta } and λ 0 ( u ) {displaystyle lambda _{0}(u)} are unknown parameters. θ = ( β , λ 0 ( u ) ) {displaystyle heta =(eta ,lambda _{0}(u))} . Here β {displaystyle eta } is finite-dimensional and is of interest; λ 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 λ 0 ( u ) {displaystyle lambda _{0}(u)} is infinite-dimensional.