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Under the partially linear model,Inserting this constant conditional contrast into the identification formula from part i gives

Semiparametric statistics studies a statistical model with a finite-dimensional target statistical parameter and an infinite-dimensional nuisance parameter. For example, the partially linear model $Y=\beta A+g(X)+\epsilon$, with $\mathbb{E}[\epsilon \mid A,X]=0$, targets the scalar $\beta$ while leaving the function $g$ and the covariate distribution unrestricted.

Writing $U=A-\mathbb{E}[A\mid X]$ and $V=Y-\mathbb{E}[Y\mid X]$ removes $g$: $V=\beta U+\epsilon$. If the needed second moments exist and $\mathbb{E}{U}^2>0$, then $\beta =\mathbb{E}[UV]/\mathbb{E}{U}^2$. The positivity condition expresses the residual variation needed to identify the target.

A nuisance tangent space describes first-order changes the nuisance can imitate. Projecting a parametric score function away from that space gives the efficient score and efficient information. Thus a finite-dimensional target can sometimes be estimated efficiently without fully specifying an infinite-dimensional distribution; the allowed model and its regularity determine whether this is possible.