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Along bounded density tilts, this functional has derivative $4\int f^{4}g=P_f(4f^{3}g)$. At a bounded baseline density its canonical gradient, relative to these regular paths, is $4(f^3-\int f^4)$. A common local bound on nearby densities suffices to obtain the same derivative along arbitrary differentiable-in-quadratic-mean paths in that bounded neighborhood. A bounded baseline alone is insufficient, because of the spike obstruction to density-power differentiability.

The mean-zero L2 space is the kernel of the bounded linear functional $h\mapsto Ph$ on L2 space for a probability measure $P$. It is a closed subspace of a Hilbert space, with inner product $P(hk)$, and is the natural ambient space for score functions and canonical gradients.

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.

For the bounded density tilt $f_t=f(1+tg)$, differentiate the polynomial in $t$:Since every score function is centered, the centered representer isA bounded baseline $f$ makes this a bounded mean-zero function, hence an element of $L_0^2(P_f)$. Thus it is the efficient influence function for the density fourth-power functional relative to the regular bounded density tilts. Its squared L2 norm is $16\{\int f^7-(\int f^4{)}^2\}$.

A bounded baseline alone does not make this functional differentiable along every quadratic-mean differentiable path. The derivative above is the intended regular-path answer. To see the need for the qualification, let $f_0=1$ and put $b(s)=6s(1-s)$ on $[0,1]$, extended by zero outside. For $0<|t|<1$, defineEach $f_t$ is a nonnegative continuous probability density function, because its narrow bump has mass $t^4$. Moreover,It is therefore a differentiable-in-quadratic-mean path with zero score function. YetThe density fourth-power functional is not even continuous along this statistical path, although every $f_t$ is individually bounded. Consequently no efficient influence function represents derivatives over the unrestricted class of all such paths.

One sufficient additional condition is a common bound $f_t,f\le M$ for all small $t$. Taylor expansion then bounds the fourth-power remainder by $6M^2(f_t-f{)}^2$, whileTogether with the quadratic-mean to L1 density derivative, this gives $\psi (P_{f_t})-\psi (P_f)=t{P}_f({\phi }_{f}g)+o(|t|)$. Under this local bound, or when the chosen statistical paths are the bounded density tilts, the boxed canonical gradient is fully justified. The counterexample is a spike obstruction to density-power differentiability.

A statistical functional is pathwise differentiable relative to chosen statistical paths if its derivative along every path depends only on that path's score function and defines a bounded linear functional on their statistical tangent space. The Riesz representation theorem expresses this derivative as an L2 inner product with a unique element of the statistical tangent space, the canonical gradient. A path family and the associated derivative remainder conditions must both be specified; a formal derivative along one convenient family does not establish differentiability along all paths.