TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Semiparametric information bound for regular asymptotically linear estimators

Statement

Assume AC and Countable Choice. Let $P$ be a law in a statistical model, let $T \subseteq L_{0}^{2} (P)$ be its closed score tangent space, and let $ψ$ be a pathwise differentiable $R^{k}$ -valued functional with canonical gradient $ϕ_{c a n} \in T^{k}$ . Write $V_{c a n} = P [ϕ_{c a n} ϕ_{c a n}^{T}]$ . Let ${\hat{ψ}}_{n}$ be asymptotically linear at $P$ with influence function $φ \in L_{0}^{2} (P)^{k}$ . Assume that for every allowed one-sided DQM path $P_{t}$ with score $s$ and every fixed $h \geq 0$ , under $P_{h / \sqrt{n}}^{\otimes n}$ the centered estimator $\sqrt{n} {{\hat{ψ}}_{n} - ψ (P_{h / \sqrt{n}})}$ converges to one law independent of $h$ , equal to its limit at $h = 0$ . Then $φ$ represents the pathwise derivative on $T$ , so

$φ \in ϕ_{c a n} + (T^{⊥})^{k}, P [φ φ^{T}] - V_{c a n} = P [(φ - ϕ_{c a n}) (φ - ϕ_{c a n})^{T}] ⪰ 0.$

By Asymptotic linearity implies asymptotic normality, this is the covariance lower bound for the estimator's asymptotic normal limit. The theorem does not assert that an estimator attaining the bound exists.

Facts & Assumptions

Given: AC and Countable Choice; the dominated model, allowed one-sided DQM paths, closed tangent space, pathwise differentiable functional, canonical gradient, asymptotic linearity, and the stated pathwise regularity under each fixed local displacement.

[F1]

Asymptotically linear estimators and influence functions: ${\hat{ψ}}_{n} - ψ (P) = (P_{n} - P) φ + o_{P} (n^{- 1 / 2})$ with $P φ = 0$ and $P ∥ φ ∥^{2} < \infty$ .

[F2]

Pathwise differentiability and parameter influence curves: for each allowed score $s$ , ${\dot{ψ}}_{P} (s)$ is the value of a continuous linear derivative on $T$ ; its canonical gradient $ϕ_{c a n} \in T^{k}$ represents that derivative.

[F3]

A one-sided DQM path yields an iid LAN proxy: for each fixed $h \geq 0$ , the one-sided DQM local alternative has a mean-one likelihood-ratio proxy, its product law is total-variation close to the actual local law, its log proxy ratio is $h n^{- 1 / 2} \sum_{i} s (X_{i}) - \frac{1}{2} h^{2} P [s^{2}] + o_{P} (1)$ , and its proxy-zero probability tends to zero.

[F4]

Le Cam's third lemma for a Gaussian likelihood-ratio limit: under a joint Gaussian limit of a statistic and the log of a mean-one likelihood-ratio proxy, the limit under the proxy law is exponentially tilted; the statistic's mean shifts by its covariance with the log ratio, while its covariance is unchanged.

[F5]

Multivariate iid central limit theorem: under AC, iid finite-second-moment vectors have the Gaussian limit with their covariance, including singular covariance.

[F6]

Slutsky's theorem for real random variables: adding a real term converging to zero in probability preserves the projected Gaussian limit.

[F7]

Cramer wold device: under AC, convergence of all real projections to the projections of a specified Borel law gives vector convergence.

[F8]

Asymptotic linearity implies asymptotic normality: the estimator's base-law limit is $N_{k} (0, Σ_{φ})$ .

[F9]

Influence-curve representers form an affine orthogonal-complement class: under Countable Choice, the ambient representers of the derivative are exactly $ϕ_{c a n} + (T^{⊥})^{k}$ .

[F10]

Cauchy-Schwarz inequality for $L^{2}$ : the $L^{2} (P)$ pairing is continuous, so equality on the score span extends to its norm closure.

[F11]

Continuous mapping theorem: the continuous map $(t, ℓ) \mapsto (t, e^{ℓ})$ preserves joint weak convergence.

[A1]

The Axiom of Choice: AC supplies the path density-version convention and is required by [F3] and [F5].

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is used by the Hilbert projection/Riesz branch that supplies the canonical gradient in [F2] and the representer decomposition in [F9].

Proof

Given: The hypotheses in the statement. Put $Σ_{φ} = P [φ φ^{T}]$ .

1.1A1F1F2F3givenalgebra

Fix an allowed path with score $s$ and a fixed $h > 0$ . Let $T_{n} = \sqrt{n} {{\hat{ψ}}_{n} - ψ (P)}$ , $Δ_{n} = n^{- 1 / 2} \sum_{i} s (X_{i})$ , and $L_{n}$ be the proxy ratio from [F3]. By [F3], $P (L_{n} = 0) \to 0$ ; set $ℓ_{n} = \log L_{n}$ when $L_{n} > 0$ and $ℓ_{n} = 0$ otherwise. For arbitrary $a \in R^{k}$ and $b \in R$ , asymptotic linearity and [F3] give $a^{T} T_{n} + b ℓ_{n} = n^{- 1 / 2} \sum_{i = 1}^{n} {a^{T} φ (X_{i}) + b h s (X_{i})} - \frac{1}{2} b h^{2} P [s^{2}] + o_{P} (1) .$ The summands are iid and square-integrable by [F1] and [F2].

2.1A1F5F6F7step 1.1given

Apply [F5] to each scalar linear combination in step 1.1, then apply [F6] to its $o_{P} (1)$ remainder. Since this holds for every $(a, b)$ , [F7] gives the joint Gaussian limit of $(T_{n}, ℓ_{n})$ , with first marginal $N_{k} (0, Σ_{φ})$ and cross-covariance between $T_{n}$ and $ℓ_{n}$ equal to $h P [φ s]$ .

3.1A1F3F4F11step 1.1step 2.1

The log-ratio marginal in step 2.1 is $N (- h^{2} P [s^{2}] / 2, h^{2} P [s^{2}])$ . Exponentiating the finite version $ℓ_{n}$ by [F11] gives the joint likelihood-ratio limit, and replacing $e^{ℓ_{n}}$ by $L_{n}$ changes it only on the $o (1)$ -probability zero event from step 1.1. Apply [F4] to the proxy law in [F3]; under that law, $T_{n} \Rightarrow N_{k} (h P [φ s], Σ_{φ})$ . Total-variation closeness in [F3] transfers the same limit to the actual law $P_{h / \sqrt{n}}^{\otimes n}$ .

4.1F2F8step 3.1given

By pathwise differentiability, $\sqrt{n} {ψ (P_{h / \sqrt{n}}) - ψ (P)} \to h {\dot{ψ}}_{P} (s)$ . The regularity hypothesis therefore makes $T_{n}$ under the actual local law converge to $N_{k} (h {\dot{ψ}}_{P} (s), Σ_{φ})$ , using the base limit [F8]. Comparing with step 3.1, uniqueness of weak limits gives $N_{k} (h P [φ s], Σ_{φ}) = N_{k} (h {\dot{ψ}}_{P} (s), Σ_{φ})$ . Equal-covariance Gaussian laws have equal means, so $P [φ s] = {\dot{ψ}}_{P} (s)$ .

5.1F2F10step 4.1

Step 4.1 holds for every allowed score. By linearity it holds on their linear span; by [F10] the pairing with $φ$ is continuous, and by [F2] the derivative is continuous, so equality extends to the closure $T$ . Thus $φ$ is an ambient representer of the pathwise derivative.

6.1A2F9step 5.1algebra∎

By [F9], $r = φ - ϕ_{c a n}$ has every coordinate in $T^{⊥}$ . Hence $P [ϕ_{c a n} r^{T}] = 0 = P [r ϕ_{c a n}^{T}]$ , and expansion gives $P [φ φ^{T}] - V_{c a n} = P [r r^{T}]$ . For every $a \in R^{k}$ , $a^{T} P [r r^{T}] a = P [(a^{T} r)^{2}] \geq 0$ , proving positive-semidefinite dominance.

Depends on

Used by

The sample mean attains the unrestricted mean bound Example

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Sources

A. W. van der Vaart, Semiparametric Statistics, Lecture Notes in Mathematics 1781, Lectures 2–4 (Springer, 2002) (standard reference, not scraped)
Edward H. Kennedy, Semiparametric theory, §§3–4 (2017) (standard reference, not scraped)
John Duchi, A few notes on contiguity, asymptotics, and local asymptotic normality, §§2 and 4.3.1 (2021) (standard reference, not scraped)