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

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Local asymptotic minimax bound

Statement

Assume AC and let $d \geq 1$ be finite. Let ${P_{n, h} : h \in R^{d}}$ be a finite-dimensional experiment that is LAN at $h = 0$ with positive-definite information $K$ and central sequence $Δ_{n} \Rightarrow N_{d} (0, K)$ as in Local asymptotic normality for local experiments. Let $M \geq 0$ be finite and $L : R^{d} \to [0, M]$ be Borel measurable, symmetric ( $L (- x) = L (x)$ ), and quasi-convex (each sublevel set is convex). Then ${lim inf}_{c \to \infty} {lim inf}_{n \to \infty} \inf_{{\hat{h}}_{n}} \sup_{∥ h ∥_{2} \leq c} E_{P_{n, h}} L ({\hat{h}}_{n} - h) \geq E L (Z), Z \sim N_{d} (0, K^{- 1}) .$ The infimum ranges over measurable, possibly randomized estimators. The experiment is pointwise LAN in each fixed $h$ ; no uniform likelihood-field assumption is added.

Facts & Assumptions

Given: AC; the pointwise LAN experiment with $K ≻ 0$ ; and the stated bounded, Borel, symmetric quasi-convex loss.

[A1]

The Axiom of Choice: AC is assumed by the Gaussian and LAN prerequisites used here.

[F1]

Local asymptotic normality for local experiments: For each fixed $h$ , there are mean-one likelihood proxies with pointwise log-likelihood expansion and total-variation approximation to the actual local law.

[F2]

Finite-prior Bayes envelopes converge under pointwise LAN: For a finite prior and a bounded Lipschitz loss, pointwise LAN makes the actual finite-prior Bayes values converge to those in the Gaussian shift.

[F3]

Finite priors approximate Gaussian-shift Bayes values: In the Gaussian shift, the Bayes value under any prior can be approximated arbitrarily closely by a finite-support prior, uniformly over decision rules.

[F4]

Bounded bowl-shaped losses have Gaussian-centered Lipschitz minorants: For every $N \geq 2$ and $m \geq 1$ it constructs a bounded Lipschitz minorant $g_{N, m} \leq L$ and proves its Gaussian shift risk is minimized at the center; it also gives $\lim_{N \to \infty} \lim_{m \to \infty} E g_{N, m} (Z) = E L (Z)$ .

[F5]

Loss functions and risk and Bayes risk and minimax risk: A rule's risk is its expected loss and its prior risk is $r_{π} (δ)$ . In this proof write $B_{E} (π, g) : = \inf_{δ} r_{π} (δ)$ for the Bayes value in experiment $E$ , with the infimum over measurable randomized estimators; this is local notation, since the supplier defines $r_{π}$ and the minimax value but does not name a Bayes-value infimum. For any prior supported in a parameter set, every rule's worst-case risk on that set is at least its prior risk. Write $B_{n}$ and $B_{G}$ for this value in the $n$ th experiment and the Gaussian shift, respectively.

[F6]

Multivariate normal law, including singular covariance, Standard normal and normal laws, The standard normal density has total mass one, and A linear map $T$ of $R^{n}$ sends Lebesgue measurable sets to Lebesgue measurable sets, with $λ_{n} (T [E]) = ∣ \det T ∣ λ_{n} (E)$ when $T$ is invertible and $T [E]$ Lebesgue null when it is not: A nondegenerate Gaussian has its usual Lebesgue density and the stated affine transforms.

[F7]

Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation: The finite Gaussian first moment is bounded by its finite second moment.

[F8]

The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ : $∥ \cdot ∥_{2}$ is the Euclidean norm.

[F9]

Borel measurable and Lebesgue measurable functions on $R^{n}$ : Borel measurability on $R^{d}$ uses the sigma-algebra generated by its open sets.

Proof

Proof technique: compare finite-prior Bayes risks, compute a Gaussian prior Bayes value, then approximate the loss from below.

1.1F4F5F6F8givenconstructalgebra

If $M = 0$ , then $L$ is identically zero and the claimed bound is equality. Assume $M > 0$ for the rest of the proof. Fix arbitrary integers $N \geq 2$ and $m \geq 1$ , and set $g = g_{N, m}$ from [F4]; it is bounded Lipschitz and $g \leq L$ . For each $ρ > 0$ , set $Γ = ρ^{- 1} K^{- 1}$ , and let $π_{Γ} = N_{d} (0, Γ)$ . Define $V_{n, c} (g) = \inf_{{\hat{h}}_{n}} \sup_{∥ h ∥_{2} \leq c} E_{P_{n, h}} g ({\hat{h}}_{n} - h)$ and $V (g) = {lim inf}_{c \to \infty} {lim inf}_{n \to \infty} V_{n, c} (g)$ . If a finite prior $π_{f}$ is supported in the radius- $c$ ball, then every worst-case risk is at least its prior average by [F5]. Taking infima gives $V_{n, c} (g) \geq B_{n} (π_{f}, g)$ , the finite-prior Bayes value.

2.1A1F4F6step 1.1givenalgebra

In the Gaussian shift, $Y ∣ h \sim N_{d} (h, K^{- 1})$ . Multiplying its likelihood by the prior density $N_{d} (0, Γ)$ and completing the square gives $H ∣ Y \sim N_{d} (S K Y, S), S = (K + Γ^{- 1})^{- 1} .$ For the fixed constructed loss $g = g_{N, m}$ , [F4] gives that the posterior risk is minimized at $S K Y$ , with value $E g (N_{d} (0, S))$ independent of $Y$ . The action $Y \mapsto S K Y$ is measurable and attains this value, so $B_{G} (π_{Γ}, g) = E g (N_{d} (0, S))$ .

2.2F1F2step 1.1givenalgebra

Let $C_{f} = \max {∥ h ∥_{2} : h \in supp (π_{f})}$ . For every $c \geq C_{f}$ , [F2] gives $B_{n} (π_{f}, g) \to B_{G} (π_{f}, g)$ , where $B_{G}$ is the Bayes value in the Gaussian shift. Hence $V (g) \geq B_{G} (π_{f}, g)$ . Since this holds for every finite prior, the bound also holds for any value approximated by finite-prior Bayes values.

3.1F3step 1.1step 2.2givenalgebra

For every $η > 0$ , [F3] supplies a finite prior $π_{f}$ whose Gaussian-shift Bayes value differs from $B_{G} (π_{Γ}, g)$ by less than $η$ . Step 2.2 therefore gives $V (g) \geq B_{G} (π_{Γ}, g) - η$ . Let $η ↓ 0$ to obtain $V (g) \geq B_{G} (π_{Γ}, g)$ .

4.1F6F7step 1.1step 2.1step 3.1algebra

Here $S = (1 + ρ)^{- 1} K^{- 1}$ . Couple $N_{d} (0, S)$ as $(1 + ρ)^{- 1 / 2} Z$ for $Z \sim N_{d} (0, K^{- 1})$ . If $B_{g}$ is a Lipschitz constant for $g$ , then $∣ E g ((1 + ρ)^{- 1 / 2} Z) - E g (Z) ∣ \leq B_{g} (1 - (1 + ρ)^{- 1 / 2}) E ∥ Z ∥_{2} \to 0$ as $ρ ↓ 0$ by [F7]. By steps 3.1 and 2.1, for each $ρ > 0$ we have $V (g) \geq E g ((1 + ρ)^{- 1 / 2} Z)$ . Since the construction applies for every $ρ > 0$ , taking $ρ ↓ 0$ yields $V (g) \geq E g (Z)$ .

5.1F4F9step 1.1step 4.1givenalgebra∎

Since $g \leq L$ , every estimator's risk under $L$ dominates its risk under this fixed $g$ , so the local minimax value for $L$ is at least $V (g)$ . Step 4.1 gives the lower bound $E g_{N, m} (Z)$ . The choices $N, m$ in step 1.1 were arbitrary; first let $m \to \infty$ and then $N \to \infty$ . By [F4] these lower bounds converge to $E L (Z)$ . This is the stated inequality.

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Sources

A. W. van der Vaart, Semiparametric Statistics, Theorem 4.14 and complete proof, pp. 46–48 (standard reference, not scraped)
John Duchi, A few notes on contiguity, asymptotics, and local asymptotic normality, Theorem 3 and §4.1 (standard reference, not scraped)