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Likelihood-ratio limit for local cones in a QMD model

Statement

Assume the Axiom of Choice (The Axiom of Choice).

Let $d \geq 1$ , let $Θ_{0} \subseteq Θ \subseteq R^{d}$ , and let $θ_{0} \in Θ_{0}$ . Assume the full-model parameter set $Θ$ is open in $R^{d}$ , so $θ_{0}$ is interior for the QMD model. Under $P_{θ_{0}}$ , let $X_{1}, \dots, X_{n}$ be IID and let $ℓ_{n} (θ) = \sum_{i = 1}^{n} \log p_{θ} (X_{i})$ be their sample log-likelihood. Define local parameter sets $C_{n} : = \sqrt{n} (Θ - θ_{0})$ and $C_{0 n} : = \sqrt{n} (Θ_{0} - θ_{0})$ . Assume the model is QMD at $θ_{0}$ as in Quadratic-mean differentiability at a parameter, and write $I : = E_{θ_{0}} [s_{θ_{0}} s_{θ_{0}}^{T}]$ for its Fisher information; assume $I$ is positive definite, meaning $v^{T} I v > 0$ for every nonzero $v$ . Suppose $C_{n}$ and $C_{0 n}$ are closed in the Euclidean metric (their complements are open), contain $0$ , and converge locally to nonempty closed convex cones $C$ and $C_{0}$ with $C_{0} \subseteq C$ . A cone contains $0$ and is closed under multiplication by every nonnegative scalar; convex means that the set contains the line segment between each pair of its points. Local convergence means that, for every $R > 0$ , there are $δ_{n, R} \to 0$ such that every point of $C_{n} \cap {\overline{B}}_{2} (0, R)$ or $C \cap {\overline{B}}_{2} (0, R)$ is within $δ_{n, R}$ in Euclidean distance of the other set, and likewise for $C_{0 n}$ and $C_{0}$ . Here a point $x$ is within $δ$ of a set $A$ when $\inf_{a \in A} ∥ x - a ∥_{2} \leq δ$ .

Assume a uniform local quadratic likelihood expansion: for $h \in C_{n}$ ,

$ℓ_{n} (θ_{0} + h / \sqrt{n}) - ℓ_{n} (θ_{0}) = h^{T} Z_{n} - \frac{1}{2} h^{T} I h + r_{n} (h),$

where $Z_{n} \Rightarrow Z \sim N_{d} (0, I)$ under $P_{θ_{0}}$ and, for every $R > 0$ , $\sup {∣ r_{n} (h) ∣ : h \in C_{n}, ∥ h ∥_{2} \leq R} \to_{P} 0$ . Assume the full and null local log-likelihoods attain maxima at ${\hat{h}}_{n} \in C_{n}$ and ${\hat{h}}_{0 n} \in C_{0 n}$ and that both maximizer sequences are tight, meaning that for every $η > 0$ some $R < \infty$ makes the limiting upper probability of either norm exceeding $R$ at most $η$ . For

$T_{n} : = 2 [\sup_{h \in C_{n}} {ℓ_{n} (θ_{0} + h / \sqrt{n}) - ℓ_{n} (θ_{0})} - \sup_{h \in C_{0 n}} {ℓ_{n} (θ_{0} + h / \sqrt{n}) - ℓ_{n} (θ_{0})}],$

the likelihood-ratio statistic satisfies

$T_{n} \Rightarrow d_{I}^{2} (I^{- 1} Z, C_{0}) - d_{I}^{2} (I^{- 1} Z, C),$

where $d_{I}$ is the Fisher-information distance of Fisher-information metric distance to a set. QMD alone is not asserted to imply the uniform expansion or tightness hypotheses.

Facts & Assumptions

Given: AC, the model, local parameter sets, expansion, convergence, and tightness hypotheses in the Statement.

[A1]

The Axiom of Choice is assumed for the finite-dimensional normal-law convention used here (The Axiom of Choice).

[F1]

QMD at an interior point means the square-root-density expansion $\int [\sqrt{p_{θ_{0} + h}} - \sqrt{p_{θ_{0}}} - \frac{1}{2} h^{T} s_{θ_{0}} \sqrt{p_{θ_{0}}}]^{2} d μ = o (∥ h ∥_{2}^{2})$ for a square-integrable score and common dominating measure (Quadratic-mean differentiability at a parameter).

[F2]

The Fisher distance is $d_{I} (x, A) = \inf_{a \in A} \sqrt{(x - a)^{T} I (x - a)}$ (Fisher-information metric distance to a set).

[F3]

The Euclidean inner product is $u^{T} v$ and its norm is $∥ u ∥_{2}$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F4]

${\overline{B}}_{2} (0, R) = {h : ∥ h ∥_{2} \leq R}$ , and the Euclidean metric has this norm as its distance (Euclidean spheres and closed balls as subspaces of $R^{n}$ ).

[F5]

The Cauchy–Schwarz inequality gives $∣ u^{T} v ∣ \leq ∥ u ∥_{2} ∥ v ∥_{2}$ (Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation).

[F6]

For nonempty closed $A$ , there is $λ > 0$ with $h^{T} I h \geq λ ∥ h ∥_{2}^{2}$ , and maximizing $q_{z} (h) = z^{T} h - \frac{1}{2} h^{T} I h$ over $A$ gives $\frac{1}{2} z^{T} I^{- 1} z - \frac{1}{2} d_{I}^{2} (I^{- 1} z, A)$ ; the maximum is attained (Quadratic likelihood optimization is squared Fisher distance).

[F7]

Under AC, $N_{d} (0, I)$ is a Borel probability law with the stated covariance (Multivariate normal law, including singular covariance).

[F8]

Weak convergence of laws gives the closed-set upper bound ${lim sup}_{n} P (Z_{n} \in F) \leq P (Z \in F)$ for each closed $F$ (Portmanteau theorem).

[F9]

Continuous maps preserve convergence in distribution at points of continuity of the limiting law (Continuous mapping theorem).

[F10]

For real random variables, adding a term converging in probability to zero preserves convergence in distribution (Slutsky's theorem for real random variables).

[F11]

Convergence in probability means that every fixed positive error threshold is exceeded with probability tending to zero (Convergence in probability); convergence in distribution is weak convergence of the laws (Convergence in distribution of random elements).

Proof

Given: The objects and hypotheses in the Statement.

1.1F1F6F3F5givenalgebra

With the supplied QMD information $I$ of [F1], write $q_{z} (h) = h^{T} z - \frac{1}{2} h^{T} I h$ and $m_{A} (z) = \sup_{h \in A} q_{z} (h)$ for $A = C, C_{0}$ . By [F6], $m_{A} (z) = \frac{1}{2} z^{T} I^{- 1} z - \frac{1}{2} d_{I}^{2} (I^{- 1} z, A)$ . If $h_{A} (z)$ is a maximizer, then $q_{z} (h_{A} (z)) \geq q_{z} (0) = 0$ . The lower bound in [F6] and [F5] imply $∥ h_{A} (z) ∥_{2} \leq 2 ∥ z ∥_{2} / λ$ (with the bound immediate when $h_{A} (z) = 0$ ).

1.2F6F3F4F5givenalgebrachoose

Let $λ$ be from [F6], fix $M > 0$ , and choose $R > 2 M / λ + 1$ . For $A_{n} = C_{n}$ , $A = C$ (and likewise for $A_{n} = C_{0 n}$ , $A = C_{0}$ ), write $m_{A_{n}, R} (z) = \sup_{h \in A_{n} \cap {\overline{B}}_{2} (0, R)} q_{z} (h)$ . For every $∥ z ∥_{2} \leq M$ , [F6] gives a maximizer $h (z)$ of $q_{z}$ on $A$ with $∥ h (z) ∥_{2} \leq 2 M / λ < R - 1$ . Local set convergence supplies points of $A_{n}$ within $δ_{n, R - 1} + 1 / n$ of $h (z)$ , hence inside the radius- $R$ ball for all large $n$ . Conversely, every point of $A_{n} \cap {\overline{B}}_{2} (0, R)$ is within $δ_{n, R} + 1 / n$ of a point of $A$ . With $B_{I} = \sum_{j, k = 1}^{d} ∣ I_{j k} ∣$ , [F3] and [F5] give $∣ q_{z} (h) - q_{z} (h^{'}) ∣ \leq (M + B_{I} (R + 1)) ∥ h - h^{'} ∥_{2}$ for $∥ h ∥_{2}, ∥ h^{'} ∥_{2} \leq R + 1$ and $∥ z ∥_{2} \leq M$ . These two directions show $\sup_{∥ z ∥_{2} \leq M} ∣ m_{A_{n}, R} (z) - m_{A} (z) ∣ \to 0$ . The same bounds apply to both full and null sets.

2.1A1F7F8step 1.1givenchoose

The sequence $Z_{n}$ is tight. Indeed, for any $η > 0$ choose $R$ with $P (∥ Z ∥_{2} \geq R) < η$ , possible because [F7] supplies a Borel probability law on finite-dimensional $R^{d}$ . The set ${z : ∥ z ∥_{2} \geq R}$ is closed, so [F8] bounds ${lim sup}_{n} P (∥ Z_{n} ∥_{2} \geq R)$ by $η$ . step 1.1 then makes the maximizers of $q_{Z_{n}}$ over $C$ and $C_{0}$ tight as well.

2.2F11step 1.2given

For $A_{n} = C_{n}$ or $C_{0 n}$ , put $L_{A_{n}, R} : = \sup_{h \in A_{n} \cap {\overline{B}}_{2} (0, R)} {ℓ_{n} (θ_{0} + h / \sqrt{n}) - ℓ_{n} (θ_{0})}$ . The uniform expansion gives $∣ L_{A_{n}, R} - m_{A_{n}, R} (Z_{n}) ∣ \leq \sup_{h \in C_{n}, ∥ h ∥_{2} \leq R} ∣ r_{n} (h) ∣ \to_{P} 0$ for each fixed $R$ . For fixed $M > 0$ and $R > 2 M / λ + 1$ , step 1.2 therefore gives, for every $ε > 0$ , $P (∣ L_{A_{n}, R} - m_{A} (Z_{n}) ∣ > ε, ∥ Z_{n} ∥_{2} \leq M) \to 0$ , with $A = C$ or $C_{0}$ , respectively.

3.1F11step 1.1step 2.2givenchoose

Given $η > 0$ , tightness of $Z_{n}$ lets us choose $M$ so that $∥ Z_{n} ∥_{2} \leq M$ with limiting probability at least $1 - η / 3$ . Choose $R > 2 M / λ + 1$ large enough that the joint event that both local likelihood maximizers lie in the radius- $R$ ball has limiting probability at least $1 - η / 3$ . By step 1.1 both quadratic maximizers also lie in that ball whenever $∥ Z_{n} ∥_{2} \leq M$ . Thus all four full and truncated suprema agree except on events with total limiting upper probability at most $2 η / 3$ . step 2.2 and arbitrariness of $η$ give $T_{n} - 2 (m_{C} (Z_{n}) - m_{C_{0}} (Z_{n})) \to_{P} 0$ .

4.1F2F6F9F10step 1.1step 1.2step 3.1∎

The form $⟨ u, v ⟩_{I} = u^{T} I v$ is an inner product; its Cauchy–Schwarz inequality follows by requiring $(u - t v)^{T} I (u - t v) \geq 0$ for every real $t$ . Thus the induced norm satisfies the triangle inequality, and taking infima gives $∣ d_{I} (x, C) - d_{I} (y, C) ∣ \leq ∥ x - y ∥_{I}$ . Also $v^{T} I v \leq B_{I} ∥ v ∥_{2}^{2}$ , where $B_{I}$ is the finite entry sum from step 1.2, so the distance map is Euclidean Lipschitz after composition with the fixed linear map $I^{- 1}$ . Therefore $g (z) : = d_{I}^{2} (I^{- 1} z, C_{0}) - d_{I}^{2} (I^{- 1} z, C)$ is continuous. The formula in step 1.1 gives $2 [m_{C} (z) - m_{C_{0}} (z)] = g (z)$ . By [F9], $g (Z_{n}) \Rightarrow g (Z)$ . step 3.1 and [F10] then give $T_{n} \Rightarrow g (Z)$ , which is the stated distance-difference law.

Remarks

The uniform local expansion and tightness are explicit hypotheses. The proof does not infer uniformity over bounded local parameters from pointwise LAN, and it does not invoke the regular interior Wilks theorem at a boundary.

Depends on

Used by

A one-sided normal likelihood ratio at the boundary Example

Dependency tree · two levels

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Sources

Peter Bartlett, Theoretical Statistics, Stat 210B Lecture 26 (standard reference, not scraped)
Steven G. Self and Kung-Yee Liang, Asymptotic Properties of Maximum Likelihood Estimators and Likelihood Ratio Tests Under Nonstandard Conditions (standard reference, not scraped)