Nuisance tangent space

Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), as required by the ambient Hilbert-space and tangent-space conventions. Let $\mathcal{M}$ be a model through $P$, let $\psi :\mathcal{M}\to {\mathbb{R}}^k$ be a specified finite-dimensional parameter-of-interest functional, and let the model designate its nuisance submodels. A nuisance submodel is a designated allowed DQM path $t\mapsto P_t$ through $P$ for which the right derivative ${\lim }_{t\downarrow 0}(\psi (P_t)-\psi (P))\mathrm{/}t$ exists and is the zero vector in ${\mathbb{R}}^k$. Its score belongs to the tangent set $\dot{\mathcal{M}}(P)$ (Tangent set of a statistical model at a law).

The nuisance tangent space ${\mathcal{T}}_{\eta }(P)$ is the norm closure in $L_0^2(P)$ of the finite real linear span of scores of nuisance submodels. It is a closed linear subspace by Closed tangent space and A tangent space is a closed linear subspace. The designated nuisance submodels are part of the model specification; this definition does not assert that every path with zero first-order target derivative is nuisance unless the model designates it as such.