Pathwise differentiability and parameter influence curves

Let $P$ be a probability law for one observation, let $\mathcal{P}$ be a model of laws containing $P$, and let $\psi :\mathcal{P}\to {\mathbb{R}}^k$ be a target functional, where $k\ge 1$ is finite. A one-sided path $t\mapsto P_t$, $0\le t<ϵ$, through $P=P_0$ is differentiable in quadratic mean (DQM) with score $s\in L^2(P)$ when its densities $p_t,p$ relative to one common sigma-finite measure $\mu$ satisfy

${\parallel \sqrt{p_t}-\sqrt{p}-\frac{t}{2}s\sqrt{p}\parallel }_{L^2(\mu )}=o(t)(t\downarrow 0)\mathrm{.}$

The score initially belongs to $L^2(P)$; A DQM score has mean zero and is unique proves it is unique modulo $P$-almost-sure equality and has mean zero, so it lies in $L_0^2(P)$ as defined by Mean-zero square-integrable functions. The allowed scores form the tangent set of the designated paths (Tangent set of a statistical model at a law); their tangent space $\mathcal{T}$ is the closure of their real linear span in $L_0^2(P)$ (Closed tangent space). It is a closed linear subspace by A tangent space is a closed linear subspace, and hence a Hilbert space under Countable Choice (The mean-zero L² space is a Hilbert space).

The functional $\psi$ is pathwise differentiable at $P$ relative to $\mathcal{T}$ if the right derivative ${\dot{\psi }}_P(s):={\lim }_{t\downarrow 0}(\psi (P_t)-\psi (P))\mathrm{/}t$ exists for every allowed path, depends only on its score, and extends to a continuous linear map ${\dot{\psi }}_P:\mathcal{T}\to {\mathbb{R}}^k$. A vector $\varphi \in L_0^2(P{)}^k$ is a parameter influence curve (or representer) when

${\dot{\psi }}_P(s)=E_P[\varphi s](s\in \mathcal{T}),$

with the expectation interpreted coordinatewise. The Hilbert-space Riesz theorem gives a unique representer in ${\mathcal{T}}^k$; it is the canonical gradient. This representation requires Countable Choice, as in Riesz representation for Hilbert spaces and The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). The supplied density versions along a path use the separate Axiom of Choice in A dominated differentiable-in-quadratic-mean submodel and The Axiom of Choice. The full set of ambient representers and its projection characterization are proved in Influence-curve representers form an affine orthogonal-complement class.

The tangent space is the closed linear span of allowed scores, not necessarily the full $L_0^2(P)$; restricting or enlarging the allowed paths can therefore change pathwise differentiability and the canonical gradient. A parameter influence curve is a derivative representer. This definition does not assert that an estimator attaining a bound exists.