Statistical functional ($\psi :\mathcal{P}\to \mathbb{R}$)

A statistical functional assigns a target value to each probability measure in a statistical model. Examples include an expected value, a quantile, and an integral of a power of a probability density function. Its behavior along statistical paths determines whether first-order influence-function representers are available.

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.

A perturbation can have negligible Hellinger distance but a large integral of its density's fourth power. At the uniform density on $[0,1]$, let $b(s)=6s(1-s)$ on $[0,1]$ and zero elsewhere, and let $f_t(u)=1-t^4+t^{-2}b(u/t^6)$. Then $\int f_t=1$ and $\int (\sqrt{f_t}-1{)}^2\le 2t^4=o(t^2)$, so the score function is zero. However $\int f_t^4\ge 72/(35t^2)$. Thus the density fourth-power functional is not continuous along this differentiable-in-quadratic-mean path, despite a bounded baseline and individually bounded nearby densities.

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.

An influence-function representer is a centered square-integrable function representing derivatives of a statistical functional along all admissible score functions. Representers may differ by a function orthogonal to the statistical tangent space. This pathwise definition is distinct from defining an influence function solely by point-mass contamination paths.

The canonical gradient is the influence-function representer in the statistical tangent space $\mathcal{T}$. It equals the orthogonal projection onto $\mathcal{T}$ of any representer. Every other representer differs from it by an element of ${\mathcal{T}}^{\perp }$, so the Pythagorean theorem in an inner-product space gives it minimum variance. For $\psi (P)=Pa$ in an unrestricted density model with bounded $a$, it is $a-Pa$.