Semiparametric Models and Tangent Spaces

Let $P$ be a probability measure on $(\mathcal{X},\mathcal{A})$. Here $L^2(P)$ is the real space of square-integrable measurable functions modulo equality $P$-almost surely, with its usual quotient norm (The space $L^p(\mu )$ as the quotient by null functions, The $L^p$ norm descends to the quotient and makes $L^p$ a normed space for $1\le p\le \mathrm{\infty }$). For $f\in L^2(P)$, the integral $Pf:=\int fdP$ is finite and independent of the representative: Cauchy–Schwarz on the probability space gives $\int \mathrm{\mid }f\mathrm{\mid }dP\le \mathrm{\parallel }f{\mathrm{\parallel }}_2$, and the expectation of integrable representatives is invariant under almost-sure equality (Cauchy-Schwarz inequality for $L^2$, Expectation depends only on the almost-everywhere class).

Define the mean-zero square-integrable space by

$L_0^2(P):=\{f\in L^2(P):Pf=0\}\mathrm{.}$

It carries the norm inherited from $L^2(P)$. Countable Choice is not needed for this definition; a later local Hilbert-space argument uses it only to inherit completeness from the library's general $L^2$ Hilbert-space result.

Let $\mathcal{M}$ be a model of probability measures on a measurable space $(\mathcal{X},\mathcal{A})$, let $ϵ>0$, and let $t\mapsto P_t\in \mathcal{M}$ be a path for $0\le t<ϵ$ with $P_0=P$. It is dominated by a common measure when one supplied sigma-finite positive measure $\mu$ satisfies $P_t\ll \mu$ for every $t$ (Dominated statistical experiments). Write $p_t=d{P}_t\mathrm{/}d\mu$ and $p=p_0$ for simultaneously supplied nonnegative density versions; the existence and normalization of each density are the Radon–Nikodym properties in Likelihood densities and A sigma-finite signed measure that is absolutely continuous with respect to a sigma-finite positive measure has a unique almost-everywhere density. The Axiom of Choice is the library convention for selecting such a family of versions along a path (The Axiom of Choice). The DQM property below is unchanged by replacing any $p_t$ by another version, since each $L^2(\mu )$ norm is unchanged by a $\mu$-null modification.

The path is differentiable in quadratic mean at $0$, with score $s\in L^2(P)$, when

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

Here $o(t)$ means that the displayed norm divided by $t$ tends to $0$ as $t\downarrow 0$; $L^2(P)$ and $L^2(\mu )$ are quotient spaces modulo almost-everywhere equality (The space $L^p(\mu )$ as the quotient by null functions). The definition is directional. A two-sided regular submodel qualifies by restricting it to $t\ge 0$. No mean-zero condition is assumed in this definition.

Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). For any probability measure $P$, the space $L_0^2(P)$ of Mean-zero square-integrable functions is a closed real linear subspace of real $L^2(P)$ with pairing $⟨f,g{⟩}_P:=\int fgdP$; consequently it is a real Hilbert space (Hilbert space). The only choice use is the Countable Choice assumption already required by the general $L^2(P)$ completeness supplier.

Assume the density-version convention for the path in A dominated differentiable-in-quadratic-mean submodel. If $s\in L^2(P)$ is a DQM score, then $Ps=0$, so $s\in L_0^2(P)$ (Mean-zero square-integrable functions). Moreover, any two scores satisfying the same DQM expansion agree $P$-almost surely. The normalization argument itself uses no choice beyond the density versions already supplied in the definition.

Fix a model $\mathcal{M}$ containing a law $P$ and designate a collection of allowed submodels through $P$. Each allowed submodel is a one-sided DQM path as in A dominated differentiable-in-quadratic-mean submodel. Its score is the $P$-almost-sure equivalence class proved unique by A DQM score has mean zero and is unique, and it belongs to $L_0^2(P)$ (Mean-zero square-integrable functions).

The tangent set $\dot{\mathcal{M}}(P)$ is the set of scores of the designated paths. The model specification determines which paths are allowed. In particular, the tangent set need not be a linear space: it may be a cone, meaning it is closed under multiplication by nonnegative real scalars, while failing closure under negative scalars. One-sided restrictions can produce this behavior. The closed linear span of the tangent set is a separate construction.

Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) so that $L_0^2(P)$ is the real Hilbert space established by The mean-zero L² space is a Hilbert space. For a tangent set $\dot{\mathcal{M}}(P)\subseteq L_0^2(P)$, its linear span is the set of all finite real linear combinations of elements of $\dot{\mathcal{M}}(P)$; the span of the empty set is $\{0\}$. Its tangent space $\mathcal{T}(P)$ is the closure of that span in the $L^2(P)$ norm. The closure consists of limits of sequences from the span, equivalently of points every positive-radius norm ball around which meets the span.

The tangent set itself is retained separately and may be a cone rather than a vector space.

Assume Countable Choice. Let $\dot{\mathcal{M}}(P)$ be any tangent set in the real Hilbert space $L_0^2(P)$. Its tangent space $\mathcal{T}(P)$, defined as the norm closure of its finite real linear span (Closed tangent space), is a closed linear subspace of $L_0^2(P)$ (Linear subspace of a vector 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.

Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), so that $L_0^2(P)$ is the real Hilbert space of The mean-zero L² space is a Hilbert space. The orthogonal complement of the nuisance tangent space within $L_0^2(P)$ is

${\mathcal{T}}_{\eta }(P{)}^{\perp }:=\{h\in L_0^2(P):⟨h,g{⟩}_P=0\text{ for every }g\in {\mathcal{T}}_{\eta }(P)\},⟨h,g{⟩}_P:=P(hg)\mathrm{.}$

This is the ordinary orthogonal complement from Orthogonality and the orthogonal complement applied in the mean-zero Hilbert space, not in all of $L^2(P)$. It is a closed linear subspace by Orthogonal complements are closed.