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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.

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.