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Convolution bound for general regular estimators

Statement

Assume AC and Countable Choice. Let $ψ$ be pathwise differentiable at $P$ with tangent space $T$ and canonical gradient $ϕ_{c a n} \in T^{k}$ , and write $V_{c a n} = P [ϕ_{c a n} ϕ_{c a n}^{T}]$ . If $T \neq {0}$ , suppose a directed family of finite-dimensional, dominated QMD models ${P_{γ, θ} : θ \in Θ_{γ} \subseteq R^{d_{γ}}}$ , with each $Θ_{γ}$ open and $P_{γ, θ_{γ, 0}} = P$ , has score spans $S_{γ} \subseteq T$ whose union is dense in $T$ . In each model let $S_{γ} \in L_{0}^{2} (P)^{d_{γ}}$ be a score basis, let $I_{γ} = P [S_{γ} S_{γ}^{T}]$ be nonsingular, and let $D_{γ} = P [ϕ_{c a n} S_{γ}^{T}]$ . Assume for each fixed $h \in R^{d_{γ}}$ that $\sqrt{n} {ψ (P_{γ, θ_{γ, 0} + h / \sqrt{n}}) - ψ (P)} \to D_{γ} h$ , and that under each local law the centered estimator $\sqrt{n} {{\hat{ψ}}_{n} - ψ (P_{γ, θ_{γ, 0} + h / \sqrt{n}})}$ converges to one common law $L$ , independent of $γ$ and $h$ . Suppose $L$ has finite covariance. By Quadratic-mean differentiability implies iid LAN, each model has the required LAN proxy; Finite-dimensional LAN convolution bound for differentiable functionals then applies with derivative matrix $D_{γ}$ . If $T = {0}$ , no submodel family is needed. Then

$Cov (L) ⪰ V_{c a n} .$

This is a lower bound for general regular estimators; asymptotic linearity is not assumed.

Facts & Assumptions

Given: AC and Countable Choice; the tangent space and canonical gradient; the directed dense family of finite-dimensional QMD submodels; a common regular limit law with finite covariance; and the finite-dimensional convolution premise stated above.

[F1]

Pathwise differentiability and parameter influence curves and Influence-curve representers form an affine orthogonal-complement class: ${\dot{ψ}}_{P}$ is continuous linear on $T$ and is represented by the canonical gradient $ϕ_{c a n} \in T^{k}$ .

[F2]

Quadratic-mean differentiability at a parameter, Quadratic-mean differentiability implies iid LAN, and Local asymptotic normality for local experiments: each finite-dimensional submodel is dominated and QMD at an interior point; for every fixed local displacement its iid experiment has the LAN proxy expansion and total-variation control required by the LAN definition.

[F3]

Regular estimator under local alternatives: regularity in a finite-dimensional local experiment means the centered estimator has a common limit law under every fixed local displacement; the statement imposes this for the displayed differentiable target in each submodel.

[F4]

Finite-dimensional LAN convolution bound for differentiable functionals: in a finite-dimensional LAN experiment with target derivative $D_{γ}$ and information $I_{γ}$ , the common regular limit law is a convolution with an independent Gaussian component of covariance $D_{γ} I_{γ}^{- 1} D_{γ}^{T}$ .

[A1]

The Axiom of Choice: AC is assumed by the QMD/LAN and finite-dimensional probability suppliers.

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice supplies the Hilbert projection and Riesz representation used for the canonical gradient.

Proof

Given: The hypotheses in the statement. For one index $γ$ , abbreviate $S = S_{γ}$ , $I = I_{γ}$ , $D = D_{γ}$ , and $S = S_{γ}$ .

1.1A2F1givenalgebra

For any $a \in R^{k}$ , the projection of $a^{T} ϕ_{c a n}$ onto $S$ is $a^{T} D I^{- 1} S$ : its residual has inner product $a^{T} D - a^{T} D I^{- 1} P [S S^{T}] = 0$ with every linear combination of the score coordinates. Therefore the covariance matrix of the projected gradient is $D I^{- 1} D^{T}$ .

2.1A1F2F3F4step 1.1given

By [F2] and Quadratic-mean differentiability implies iid LAN, this finite-dimensional submodel has the required LAN proxy and total-variation control. The regularity premise in the statement and [F4] give $L \overset{d}{=} G_{γ} + W_{γ}$ , where $G_{γ} \sim N_{k} (0, D I^{- 1} D^{T})$ is independent of $W_{γ}$ . Since $L$ has finite covariance and $G_{γ}$ does, $W_{γ}$ has finite covariance as well; independence yields $Cov (L) = D I^{- 1} D^{T} + Cov (W_{γ}) ⪰ D I^{- 1} D^{T}$ .

2.2F1step 1.1givenchoosealgebra

If $T = {0}$ , then $ϕ_{c a n} = 0$ and $Cov (L) ⪰ 0 = V_{c a n}$ . Otherwise fix $a \in R^{k}$ and $ε > 0$ . Density of the union of the $S_{γ}$ in $T$ provides an index $γ$ and $g \in S_{γ}$ with $∥ g - a^{T} ϕ_{c a n} ∥_{L^{2} (P)} < ε$ . The projection formula in step 1.1 is orthogonal, so for any $g \in S_{γ}$ , $∥ a^{T} ϕ_{c a n} - g ∥_{2}^{2} = ∥ a^{T} ϕ_{c a n} - Π_{S_{γ}} (a^{T} ϕ_{c a n}) ∥_{2}^{2} + ∥ Π_{S_{γ}} (a^{T} ϕ_{c a n}) - g ∥_{2}^{2}$ . Thus the projection error is below $ε$ , and the quadratic forms $a^{T} D_{γ} I_{γ}^{- 1} D_{γ}^{T} a$ approach $a^{T} V_{c a n} a$ as $ε ↓ 0$ .

3.1step 2.1step 2.2algebra∎

When $T \neq {0}$ , step 2.1 gives $a^{T} Cov (L) a \geq a^{T} D_{γ} I_{γ}^{- 1} D_{γ}^{T} a$ for every $γ$ . Take the approximation in step 2.2 with arbitrarily small $ε$ to obtain $a^{T} Cov (L) a \geq a^{T} V_{c a n} a$ . Since this holds for every $a$ , $Cov (L) ⪰ V_{c a n}$ . Together with the zero-tangent case in step 2.2, this proves the claim.

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Sources

A. W. van der Vaart, Semiparametric Statistics, Lecture Notes in Mathematics 1781, Lectures 2 and 4 (Springer, 2002) (standard reference, not scraped)
Edward H. Kennedy, Semiparametric theory, §§3–5 (2017) (standard reference, not scraped)