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Asymptotic linearity and normality of iid Z-estimators

Statement

Assume the Axiom of Choice (AC), as required by the library's multivariate iid central limit theorem. Let $(X, A)$ be a measurable space, let $Θ \subset R^{d}$ be open with its Borel sigma-algebra, and let $θ_{0} \in Θ$ . Let $X_{1}, X_{2}, \dots$ be iid $X$ -valued observations with law $P$ . Let $ψ : Θ \times X \to R^{d}$ be jointly measurable and satisfy $P ∥ ψ_{θ} ∥ < \infty$ for every $θ \in Θ$ ; write $ψ_{θ} (x) = ψ (θ, x)$ . Let ${\hat{θ}}_{n} : (X^{n}, A^{\otimes n}) \to (Θ, B (Θ))$ be a measurable estimator sequence as in Estimator sequences in IID experiments and a Z-estimator as in Z-estimators, with the stronger root-rate condition below. Write $P ψ_{θ} = E [ψ_{θ} (X_{1})], P_{n} ψ_{θ} = \frac{1}{n} \sum_{i = 1}^{n} ψ_{θ} (X_{i}), G_{n} ψ_{θ} = \sqrt{n} (P_{n} - P) ψ_{θ} .$ Assume the displayed random vectors and suprema below are measurable, and suppose:

$∥ {\hat{θ}}_{n} - θ_{0} ∥ = o_{P} (1)$ and $\sqrt{n} ∥ P_{n} ψ_{{\hat{θ}}_{n}} ∥ = o_{P} (1)$ .
$P ψ_{θ_{0}} = 0$ , $P ∥ ψ_{θ_{0}} ∥^{2} < \infty$ , and $Σ = P [ψ_{θ_{0}} ψ_{θ_{0}}^{T}]$ .
The mean map $Ψ (θ) = P ψ_{θ}$ has expansion $Ψ (θ_{0} + h) = V h + ρ (h)$ with $V$ nonsingular and $∥ ρ (h) ∥ / ∥ h ∥ \to 0$ as $h \to 0$ , $h \neq 0$ .
The local stochastic-equicontinuity condition holds: for every $ε > 0$ , $\lim_{δ ↓ 0} {lim sup}_{n \to \infty} P^{n} ⁣ (\sup_{θ \in Θ : ∥ θ - θ_{0} ∥ < δ} ∥ G_{n} (ψ_{θ} - ψ_{θ_{0}}) ∥ > ε) = 0.$

Here $P^{n}$ denotes the product sampling law of $(X_{1}, \dots, X_{n})$ .

Then $\sqrt{n} ({\hat{θ}}_{n} - θ_{0}) = - V^{- 1} G_{n} ψ_{θ_{0}} + o_{P} (1) \Rightarrow N_{d} (0, V^{- 1} Σ V^{- T}) .$ This theorem assumes consistency and the local empirical-process condition; it asserts neither. The local equicontinuity condition is the direct process hypothesis used in the proof, and consistency alone does not replace it.

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[A1]

$∥ {\hat{θ}}_{n} - θ_{0} ∥ = o_{P} (1)$ and $\sqrt{n} ∥ P_{n} ψ_{{\hat{θ}}_{n}} ∥ = o_{P} (1)$ .

[A2]

For every $ε > 0$ , the local stochastic-equicontinuity limit in condition 4 of the statement is zero.

[A3]

$Ψ (θ_{0} + h) = V h + ρ (h)$ , $V$ is nonsingular, and $∥ ρ (h) ∥ / ∥ h ∥ \to 0$ .

[A4]

$P ψ_{θ_{0}} = 0$ and $P ∥ ψ_{θ_{0}} ∥^{2} < \infty$ .

[F1]

Stochastic order and convergence notation for finite-dimensional random vectors: $O_{P} (1)$ and $o_{P} (1)$ are defined by boundedness and convergence of tail probabilities.

[F2]

Finite-dimensional stochastic-order calculus and tightness: weak convergence implies finite-dimensional tightness, and $o_{P} (1) O_{P} (1) = o_{P} (1)$ .

[F3]

Vector Slutsky theorem: joint convergence with a probability-limit constant and continuous finite-dimensional transformations require no independence.

[F4]

Convergence in distribution of random elements: convergence in distribution means weak convergence of laws, with no common probability space required.

[F5]

Identical distribution and IID families: iid means a common law and mutual independence of the whole family.

[F6]

The real Frechet derivative on real and complex matrix spaces with the Frobenius norm: a finite-dimensional derivative is the real-linear first-order approximation with remainder $o (∥ h ∥)$ .

[F7]

Multivariate iid central limit theorem: under AC, iid finite-dimensional vectors with finite second moment have a centered normal limit with their covariance; singular covariance is allowed.

[F8]

Multivariate normal law, including singular covariance: under AC, $N_{d} (0, Σ)$ is characterized by all scalar projections having their corresponding centered normal laws; its covariance is $Σ$ .

[F9]

The Axiom of Choice: every family of nonempty sets has a choice function.

[F10]

Z-estimators: a measurable estimator sequence is a Z-estimator when it is an exact root or an $o_{P} (1)$ -approximate root of its empirical equation.

[F11]

Estimator sequences in IID experiments: an estimator sequence consists of measurable maps from each product sample space $(X^{n}, A^{\otimes n})$ into its Borel target space.

[F12]

Measurable coordinatewise functions preserve independence: applying measurable coordinatewise maps to an independent family preserves independence.

Proof

Given: AC, the iid observations and score functions, and assumptions [A1]–[A4].

1.1A1A2choose

Fix $ε, η > 0$ . By [A2] choose $δ > 0$ such that the limiting upper probability of the displayed supremum exceeding $ε$ is at most $η$ . By [A1], $P^{n} (∥ {\hat{θ}}_{n} - θ_{0} ∥ \geq δ) \to 0$ . On the complementary event, $∥ G_{n} (ψ_{{\hat{θ}}_{n}} - ψ_{θ_{0}}) ∥$ is bounded by that supremum. Hence its limiting upper tail probability is at most $η$ ; letting $η ↓ 0$ proves $G_{n} (ψ_{{\hat{θ}}_{n}} - ψ_{θ_{0}}) = o_{P} (1)$ .

1.2A4F2F4F5F7F9F12

By [F12], applying the same measurable map $ψ_{θ_{0}}$ to the iid observations preserves independence; identical distribution follows because every $X_{i}$ has the same law. By [A4] and [F5], $G_{n} ψ_{θ_{0}} = n^{- 1 / 2} \sum_{i = 1}^{n} ψ_{θ_{0}} (X_{i})$ is a centered iid sum with finite covariance $Σ$ . AC is available by [F9], so [F7] gives $G_{n} ψ_{θ_{0}} \Rightarrow N_{d} (0, Σ)$ , where convergence in distribution is weak convergence of laws by [F4]. In particular $G_{n} ψ_{θ_{0}} = O_{P} (1)$ by [F2].

2.1A1A3F6F10F11step 1.1algebra

Put $h_{n} = {\hat{θ}}_{n} - θ_{0}$ and $W_{n} = \sqrt{n} h_{n}$ . By [F10]–[F11], the estimator and root have the stated Z-estimator and data-measurability conventions; [A1] imposes the stronger root-rate residual, which is also $o_{P} (1)$ since $n^{- 1 / 2} \leq 1$ . The identity $\sqrt{n} P_{n} ψ_{{\hat{θ}}_{n}} = G_{n} ψ_{{\hat{θ}}_{n}} + \sqrt{n} Ψ ({\hat{θ}}_{n})$ , [A1], step 1.1 and the derivative expansion in [A3], as specified by [F6], give $V W_{n} + \sqrt{n} ρ (h_{n}) = - G_{n} ψ_{θ_{0}} + o_{P} (1)$ . Moreover $∥ ρ (h_{n}) ∥ / ∥ h_{n} ∥ = o_{P} (1)$ by [A1] and [A3], with the ratio set to zero when $h_{n} = 0$ .

3.1A3F2step 1.2step 2.1algebra

Since $V$ is nonsingular, [A3] and step 2.1 imply $∥ W_{n} ∥ \leq ∥ V^{- 1} ∥ (∥ G_{n} ψ_{θ_{0}} ∥ + e_{n} + a_{n} ∥ W_{n} ∥)$ , where $e_{n} = o_{P} (1)$ and $a_{n} = ∥ ρ (h_{n}) ∥ / ∥ h_{n} ∥ = o_{P} (1)$ (set $a_{n} = 0$ if $h_{n} = 0$ ). With probability tending to one, $∥ V^{- 1} ∥ a_{n} \leq 1 / 2$ . Absorbing that term and using $G_{n} ψ_{θ_{0}} = O_{P} (1)$ from step 1.2 proves $W_{n} = O_{P} (1)$ .

4.1A3F1F2step 2.1step 3.1algebra

Now $\sqrt{n} ρ (h_{n}) = o_{P} (1)$ because it is the product of $∥ W_{n} ∥ = O_{P} (1)$ and $∥ ρ (h_{n}) ∥ / ∥ h_{n} ∥ = o_{P} (1)$ . Substituting into step 2.1 gives $W_{n} = - V^{- 1} G_{n} ψ_{θ_{0}} + o_{P} (1)$ .

5.1F3F4F7F8F9step 1.2step 4.1∎

By [F4], [F7], and [F8], the continuous linear image $- V^{- 1} G_{n} ψ_{θ_{0}}$ converges to a centered Gaussian with covariance $V^{- 1} Σ V^{- T}$ : for every $u \in R^{d}$ , the projection of $N_{d} (0, Σ)$ along $- V^{- T} u$ is centered normal with variance $u^{T} V^{- 1} Σ V^{- T} u$ . Applying [F3] to the $o_{P} (1)$ remainder in step 4.1 proves the asserted normal limit. AC is used in the multivariate iid CLT and the normal-law supplier, not in the stochastic-order or Taylor steps.

Remarks

By Z-estimators, an $o_{P} (1)$ root residual is enough for the Z-estimator definition; condition 1 strengthens it to $o_{P} (n^{- 1 / 2})$ . The theorem assumes consistency and local stochastic equicontinuity; it does not use Consistency of Z-estimators under a uniform law or infer equicontinuity from consistency alone. The proof uses the quantified process condition directly, without a uniform-law theorem.

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Sources

Peter Bartlett, Theoretical Statistics, Stat 210B Lecture 16, PDF p. 13 (standard reference, not scraped)
Aad W. van der Vaart, Semiparametric Statistics, Lecture 6 §6.5, Theorem 6.19, PDF pp. 73–74 (standard reference, not scraped)