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Quadratic-mean differentiability implies iid LAN

Statement

Assume the Axiom of Choice. Let ${P_{θ} : θ \in Θ \subseteq R^{d}}$ be dominated by a common sigma-finite measure, with $Θ$ open, and suppose the model is QMD at an interior $θ_{0}$ with score $s$ and information $I = E_{θ_{0}} [s s^{T}]$ . Let $X_{1}, \dots, X_{n}$ be IID $P_{θ_{0}}$ , and for each fixed local sequence $h_{n} \to h$ set $u_{n} = h_{n} / \sqrt{n}$ . Then the local product laws $P_{θ_{0} + u_{n}}^{\otimes n}$ admit mean-one likelihood-ratio proxies $L_{n}$ relative to $P_{θ_{0}}^{\otimes n}$ such that ${∥ P_{θ_{0} + u_{n}}^{\otimes n} - L_{n} P_{θ_{0}}^{\otimes n} ∥}_{T V} \to 0,$ and under $P_{θ_{0}}^{\otimes n}$ , $\log L_{n} = h_{n}^{T} Δ_{n} - \frac{1}{2} h^{T} I h + o_{P} (1), Δ_{n} = n^{- 1 / 2} \sum_{i = 1}^{n} s (X_{i}) \Rightarrow N_{d} (0, I) .$ Thus the IID QMD experiment is LAN in the proxy sense of Local asymptotic normality for local experiments. This assertion does not claim that the full local product law has a Radon–Nikodym derivative relative to the base law.

Facts & Assumptions

Given: AC; the QMD model, common dominator, score, and information above; and IID observations from $P_{θ_{0}}$ .

[A1]

The Axiom of Choice: AC is the stated set-theoretic axiom. It is assumed here because the multivariate IID CLT supplier explicitly assumes it.

[L1]

Likelihood densities: The chosen density versions satisfy $P_{θ} (A) = \int_{A} p_{θ} d μ$ for every measurable $A$ .

[F1]

Quadratic-mean differentiability at a parameter: At $θ_{0}$ , the square-root densities have an $L^{2} (μ)$ first-order expansion with score $s$ .

[F2]

Normalized product likelihood expansion under QMD: For every $h_{n} \to h$ , the normalized absolutely continuous product proxy has mean-one likelihood $L_{n}$ , differs from the full product law in total variation by $o (1)$ , has $P_{θ_{0}}^{\otimes n} (L_{n} = 0) \to 0$ , and its log likelihood has expansion $h_{n}^{T} Δ_{n} - \frac{1}{2} h^{T} I h + o_{P} (1)$ .

[F3]

Multivariate iid central limit theorem: Under AC and for a finite integer $d \geq 1$ , IID vectors with finite second moment and covariance $I$ satisfy $n^{- 1 / 2} \sum_{i} (X_{i} - E X_{i}) \Rightarrow N_{d} (0, I)$ ; singular $I$ is allowed.

[F4]

Local asymptotic normality for local experiments: LAN means a Gaussian central-sequence limit, the stated log-likelihood expansion for each fixed local direction, and a total-variation-close mean-one absolutely continuous proxy.

[F5]

The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ : The local directions and score sums use the Euclidean inner product and norm on $R^{d}$ .

[F6]

A QMD score is centered and has finite information: A QMD score has mean zero and finite positive-semidefinite information matrix.

Proof

Proof technique: combine the normalized QMD likelihood expansion with the score central limit theorem.

1.1F4F5givenconstructalgebra

If $d = 0$ , the parameter space containing $θ_{0}$ is the singleton empty tuple by [F5]. Thus $h_{n} = h = 0$ , $u_{n} = 0$ , and every local law equals the base law. The score and $Δ_{n}$ are empty vectors and $I$ is the empty matrix. Choose $L_{n} = 1$ : its mean is one, the total variation error is zero, and the log expansion is the exact identity $0 = 0$ . The law of $Δ_{n}$ is the unique law on the singleton, denoted $N_{0} (0, 0)$ under the empty-tuple convention; its weak convergence is immediate. These facts give LAN by [F4] without a CLT. Henceforth suppose $d \geq 1$ .

2.1A1F1F2L1step 1.1given

By [A1, L1, F1] the path is QMD with the specified density versions. [F2] applies to the chosen sequence $u_{n} = h_{n} / \sqrt{n}$ and supplies the normalized proxy $L_{n}$ , the total variation approximation, and the log expansion. Its normalizer is $1 - α_{n} > 0$ for all sufficiently large $n$ , since $α_{n} = o (∥ u_{n} ∥_{2}^{2})$ and $u_{n} \to 0$ .

2.2A1F1F3F5F6step 1.1given

By [F6], the score has $E_{θ_{0}} s = 0$ and covariance $E_{θ_{0}} [s s^{T}] = I$ with finite entries. QMD also supplies $E_{θ_{0}} ∥ s ∥_{2}^{2} < \infty$ by [F1]. Thus [F3] applies to $s (X_{i})$ and yields $Δ_{n} \Rightarrow N_{d} (0, I)$ , including singular information.

3.1F2F4step 1.1step 2.1step 2.2∎

For each fixed $h$ , steps 2.1 and 2.2 give the proxy approximation, the log expansion and the Gaussian central-sequence limit required in [F4]. Hence the local experiment is LAN. The proof used the full likelihood only for the normalized absolutely continuous component; singular mass is handled by the total-variation bound in [F2]. This proves the stated proxy-LAN conclusion.

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Sources

Peter Bartlett, Theoretical Statistics, Stat 210B Lecture 22, PDF pp. 8–11 (standard reference, not scraped)
John Duchi, A few notes on contiguity, asymptotics, and local asymptotic normality, §2 (standard reference, not scraped)