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For a pathwise differentiable statistical functional, an influence-function representer is a mean-zero function $\phi \in L^2(P)$ such that every admissible score function $g$ satisfies $D{\psi }_P(g)=P(\phi g)$. The efficient influence function, also called the canonical gradient, is the unique such representer in the statistical tangent space. Equivalently, it is the orthogonal projection of any representer onto that statistical tangent space. The Pythagorean theorem in an inner-product space shows that it has the smallest squared L2 norm among all representers.

Here the statistical tangent space is all of $L_0^2(P_f)$. To verify the closure explicitly, take $h\in L_0^2(P_f)$, truncate it to $h_m=\max (-m,\min (h,m))$, and set $g_m=h_m-P_f{h}_m$. Then $g_m$ is bounded and centered, and $g_m\to h$ in $L^2(P_f)$, by dominated convergence and the Cauchy-Schwarz inequality. Part (c) supplies a representer already in this space. HenceIts variance is $P_f{a}^2-(P_{f}a{)}^2$.

The closure must be taken in the density-weighted space $L^2(P_f)$. An unweighted reading of $L^2[0,1]$ in the printed hint is false. For example, when $f(u)=2u$, the function $h(u)=u^{-3/4}-8/5$ has $P_{f}h=0$ and $P_f{h}^2=36/25$, so bounded centered truncations converge to it in $L^2(P_f)$; nevertheless $h\notin L^2([0,1],du)$. This illustrates density of bounded centered scores and fixes the measure in the closure statement.

Vary only the probability density function of $X$, using ${\eta }_t=\eta (1+tb)$ with bounded $b$ and $E_{\eta }b(X)=0$. The nuisance score function is $b(X)$, giving the statistical tangent setIts closed linear span is the nuisance tangent space of all centered $L^2(\eta )$ functions of $X$, by density of bounded centered scores.

The conditional expectation of the parametric score function given $X$ is zero:It is therefore orthogonal to this nuisance tangent space. Its orthogonal projection onto that space vanishes, so the efficient score is unchanged. By independence and $E{\epsilon }^2={\sigma }^2$,These equal the parametric score function and Fisher information when $\eta$ is known. There is no loss of information from the unknown covariate density. This is adaptivity to an unknown covariate distribution; it follows from score orthogonality, without having to estimate the nuisance parameter.