= Semiparametric statistics
{wiki}
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> $Y=\beta A+g(X)+\varepsilon$, with $\mathbb E[\varepsilon\mid A,X]=0$, 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 $g$: $V=\beta U+\varepsilon$. If the needed second moments exist and $\mathbb EU^2>0$, then $\beta=\mathbb E[UV]/\mathbb EU^2$. 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.
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