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A consistent approximate MLE has the misspecified sandwich limit under local LAN conditions

Statement

Assume the hypotheses of A local KL expansion and square-integrable log-Lipschitz envelope give misspecified iid LAN and AC. Let ${\hat{θ}}_{n}$ be a measurable estimator such that ${\hat{θ}}_{n} \to θ^{*}$ in $P_{0}$ -probability and $P_{n} \log p_{{\hat{θ}}_{n}} \geq \sup_{θ \in Θ} P_{n} \log p_{θ} - o_{P_{0}} (n^{- 1}),$ where the empirical supremum is finite and measurable and the approximation remainder is measurable. Set $P_{n} f = n^{- 1} \sum_{i = 1}^{n} f (X_{i})$ . Then, with $I = E_{0} [s s^{T}]$ and $S_{n} = n^{- 1 / 2} \sum_{i} s (X_{i})$ , $\sqrt{n} ({\hat{θ}}_{n} - θ^{*}) = V^{- 1} S_{n} + o_{P_{0}} (1) \Rightarrow N_{d} (0, V^{- 1} I V^{- 1}) .$ Thus the MLE sampling covariance is the sandwich matrix $V^{- 1} I V^{- 1}$ , whereas the BvM posterior covariance is $V^{- 1}$ . For the optional recentring conclusion, suppose the posterior $Π_{n}$ is formed from a proper prior with a Lebesgue density continuous and strictly positive on a neighbourhood of $θ^{*}$ , its evidence is finite and positive almost surely, and for every deterministic sequence $M_{n} \to \infty$ , $Π_{n} ⁣ (∥ θ - θ^{*} ∥_{2} > n^{- 1 / 2} M_{n} ∣ X_{1}, \dots, X_{n}) = o_{P_{0}} (1) .$ Then ${∥ Π_{n} ⁣ (\sqrt{n} (θ - θ^{*}) \in \cdot ∣ X_{1}, \dots, X_{n}) - N_{d} ⁣ (\sqrt{n} ({\hat{θ}}_{n} - θ^{*}), V^{- 1}) ∥}_{T V} = o_{P_{0}} (1),$ by Compact-uniform LAN and rate contraction imply a total-variation BvM limit and the estimator expansion proved here. The consistency and approximate-maximization hypotheses are separate assumptions; this theorem proves neither MLE consistency nor existence of an MLE.

Facts & Assumptions

Given: The iid LAN hypotheses and notation, AC, a measurable consistent estimator, its stated $o_{P_{0}} (n^{- 1})$ approximate-maximization property, and, for the optional recentering conclusion, a proper prior with a Lebesgue density continuous and strictly positive near $θ^{*}$ , finite positive evidence almost surely, and posterior contraction outside $n^{- 1 / 2} M_{n}$ for every deterministic $M_{n} \to \infty$ .

[F1]

A local KL expansion and square-integrable log-Lipschitz envelope give misspecified iid LAN: The score has mean zero and finite second moment, and the log-likelihood expansion is uniform on each compact set of local parameters.

[F2]

Stochastic order and convergence notation for finite-dimensional random vectors: $o_{P} (1)$ denotes convergence in probability to zero and $O_{P} (1)$ denotes boundedness in probability.

[F3]

A square-integrable Lipschitz envelope controls a local empirical process: A local log-likelihood class with square-integrable Lipschitz envelope has expected centered-process supremum at most $C_{d} r (E_{0} M^{2})^{1 / 2}$ on a radius- $r$ ball.

[F4]

Multivariate iid central limit theorem: Under AC, iid finite-second-moment vectors satisfy the multivariate CLT, including singular covariance.

[F5]

Finite-dimensional stochastic-order calculus and tightness: Finite-dimensional convergence in distribution implies boundedness in probability.

[F6]

Vector Slutsky theorem: If $Y_{n} - c = o_{P} (1)$ and $X_{n} \Rightarrow X$ , then $(X_{n}, Y_{n}) \Rightarrow (X, c)$ .

[F7]

Total variation distance for probability laws: $∥ P - Q ∥_{T V} = \sup_{A} ∣ P (A) - Q (A) ∣$ .

[F8]

Multivariate normal law, including singular covariance, A positive-definite multivariate normal law has its Gaussian density: Under AC, the multivariate-normal law exists for every positive-semidefinite covariance, including a possibly singular sandwich limit; the positive-definite Gaussian with covariance $V^{- 1}$ also has its usual Lebesgue density for the optional recentering argument.

[F9]

Compact-uniform LAN and rate contraction imply a total-variation BvM limit: For a dominated finite-dimensional experiment with supplied jointly measurable density versions, a proper Lebesgue prior density continuous and strictly positive near the interior center, finite positive evidence almost surely, compact-uniform LAN, and contraction outside $δ_{n} M_{n}$ for every deterministic $M_{n} \to \infty$ , the posterior has the stated total-variation BvM limit.

[A1]

The Axiom of Choice: AC is used by the CLT [F4] and Gaussian-law/density suppliers [F8], and nowhere in the likelihood-optimization argument.

Proof

technique · First establish root-$n$ tightness by testing dyadic shells against the local quadratic KL loss; then use the compact-uniform LAN quadratic to identify the approximate maximizer

Given: The objects and hypotheses in the Statement.

1.1F1F3givenchoose

Write $f_{t} = \log (p_{θ^{*} + t} / p_{θ^{*}})$ . Positive definiteness of $V$ and the contrast expansion in [F1] give constants $c, ρ > 0$ with $E_{0} f_{t} \leq - c ∥ t ∥_{2}^{2}$ for $∥ t ∥_{2} \leq ρ$ . Use the measurable versions from [F3], obtained by zero extension off the common log-Lipschitz full set. The log-Lipschitz envelope and [F3] give $E_{0} \sup_{∥ t ∥ \leq r} ∣ G_{n} f_{t} ∣ \leq C r$ for $r \leq ρ$ , where $G_{n} = \sqrt{n} (P_{n} - P_{0})$ and $C = C_{d} (E_{0} M^{2})^{1 / 2}$ .

2.1F1F2F3step 1.1givenchoosealgebra

Let $η_{n} = n (\sup_{θ \in Θ} P_{n} \log p_{θ} - P_{n} \log p_{{\hat{θ}}_{n}})$ . The assumption gives $η_{n} = o_{P_{0}} (1)$ by [F2]. For fixed $A \geq 1$ , set $r_{j} = 2^{j} A / \sqrt{n}$ and consider shells $r_{j} \leq ∥ {\hat{θ}}_{n} - θ^{*} ∥_{2} \leq 2 r_{j}$ with $2 r_{j} \leq ρ$ . On such a shell, approximate maximization implies $n P_{n} f_{{\hat{θ}}_{n} - θ^{*}} \geq - η_{n}$ , while step 1.1 bounds the mean term above by $- c n r_{j}^{2}$ . On ${η_{n} \leq c A^{2} / 2}$ this forces $\sup_{∥ t ∥ \leq 2 r_{j}} ∣ G_{n} f_{t} ∣ \geq (c / 2) \sqrt{n} r_{j}^{2}$ . For nonnegative $Z$ and $a > 0$ , $a 1_{{Z \geq a}} \leq Z$ implies $P_{0} (Z \geq a) \leq E_{0} Z / a$ ; apply this to the supremum and threshold above, then step 1.1 gives $C^{'} / (\sqrt{n} r_{j}) = C^{'} / (2^{j} A)$ . Summing over the shells, using consistency to remove $∥ {\hat{θ}}_{n} - θ^{*} ∥ > ρ / 4$ , and then taking $A \to \infty$ proves $\sqrt{n} ({\hat{θ}}_{n} - θ^{*}) = O_{P_{0}} (1)$ .

3.1A1F4F5step 2.1given

Set ${\hat{h}}_{n} = \sqrt{n} ({\hat{θ}}_{n} - θ^{*})$ and $h_{n}^{*} = V^{- 1} S_{n}$ . By [F4] the score sums converge in distribution; [F5] makes $S_{n}$ and $h_{n}^{*}$ bounded in probability. Step 2.1 gives the same for ${\hat{h}}_{n}$ . Since $θ^{*}$ is interior, $θ^{*} + h_{n}^{*} / \sqrt{n} \in Θ$ with probability tending to one.

4.1F1step 3.1givenalgebra

On any fixed ball containing ${\hat{h}}_{n}$ and $h_{n}^{*}$ with probability arbitrarily close to one, [F1] expands both log likelihoods uniformly. Approximate maximization then gives $Q_{n} ({\hat{h}}_{n}) \geq Q_{n} (h_{n}^{*}) - o_{P_{0}} (1), Q_{n} (h) = h^{T} S_{n} - \frac{1}{2} h^{T} V h .$ Completing the square gives $Q_{n} (h_{n}^{*}) - Q_{n} ({\hat{h}}_{n}) = \frac{1}{2} ({\hat{h}}_{n} - h_{n}^{*})^{T} V ({\hat{h}}_{n} - h_{n}^{*}) \geq \frac{1}{2} λ_{\min} (V) ∥ {\hat{h}}_{n} - h_{n}^{*} ∥_{2}^{2}$ . Hence ${\hat{h}}_{n} - h_{n}^{*} = o_{P_{0}} (1)$ .

5.1A1F1F4F6F8step 4.1

By [F1], $V^{- 1} s (X_{i})$ is centered, iid, and has covariance $V^{- 1} I V^{- 1}$ . Applying the multivariate CLT [F4] to these vectors gives $h_{n}^{*} = V^{- 1} S_{n} \Rightarrow N_{d} (0, V^{- 1} I V^{- 1})$ , including when this covariance is singular. Then [F6] and step 4.1 give ${\hat{h}}_{n} \Rightarrow N_{d} (0, V^{- 1} I V^{- 1})$ . This is the sandwich sampling covariance; it need not equal the posterior covariance $V^{- 1}$ .

6.1F7F8F9step 4.1givenalgebra∎

For each $n$ , the iid sample experiment is dominated by the finite product $μ^{\otimes n}$ , with densities $p_{0, n} (x_{1}, \dots, x_{n}) = \prod_{i} p_{0} (x_{i})$ and $p_{n, θ} (x_{1}, \dots, x_{n}) = \prod_{i} p_{θ} (x_{i})$ . Iterated integration gives the product-law formula on measurable rectangles, so uniqueness of product measures identifies these as densities of $P_{0}^{\otimes n}$ and $P_{θ}^{\otimes n}$ . Their likelihood ratio is the sum in the LAN conclusion of [F1]. Under the optional prior, evidence, and posterior-rate conditions in the Statement, [F9] therefore gives the BvM approximation centered at $h_{n}^{*}$ . Let $φ_{V}$ be the density of $N_{d} (0, V^{- 1})$ from [F8]. The fundamental theorem of calculus and translation invariance of Lebesgue measure give $\int ∣ φ_{V} (x - a) - φ_{V} (x - b) ∣ d x \leq ∥ a - b ∥_{2} \int ∥ \nabla φ_{V} (x) ∥_{2} d x .$ The last integral is finite because $\nabla φ_{V} (x) = - V x φ_{V} (x)$ and the Gaussian has a finite first moment. Taking $a = {\hat{h}}_{n}$ , $b = h_{n}^{*}$ and using step 4.1 makes this Gaussian-shift TV distance $o_{P_{0}} (1)$ . The triangle inequality with the BvM approximation proves the recentered conclusion.

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Sources

B. J. K. Kleijn and A. W. van der Vaart, The Bernstein-Von-Mises theorem under misspecification, Lemma 2.2 and proof conditions (standard reference, not scraped)
Peter Bartlett, Theoretical Statistics, Lecture 23, maximum likelihood conditions under QMD (standard reference, not scraped)