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Consistency of Z-estimators under a uniform law

Statement

Let $(Θ, d)$ be a metric space, let $X_{1}, \dots, X_{n}$ be IID with law $P$ , and let $ψ_{θ}$ be jointly measurable and integrable as an $R^{q}$ -valued function for a fixed finite $q \geq 1$ . Write $Ψ_{n} (θ) = P_{n} ψ_{θ}$ and $Ψ (θ) = P ψ_{θ}$ , using a fixed norm on $R^{q}$ . Fix $θ_{0} \in Θ$ . Assume the displayed suprema and estimator evaluations are measurable. If

$\sup_{θ \in Θ} ∥ Ψ_{n} (θ) - Ψ (θ) ∥ = o_{P} (1)$ ;
$Ψ (θ_{0}) = 0$ and, for every $ε > 0$ , $b_{ε} = \inf {∥ Ψ (θ) ∥ : d (θ, θ_{0}) \geq ε} > 0$ , with $\inf \emptyset = + \infty$ ;
$∥ Ψ_{n} ({\hat{θ}}_{n}) ∥ = o_{P} (1)$ for a measurable estimator ${\hat{θ}}_{n}$ ,

then ${\hat{θ}}_{n}$ is consistent in probability for $θ_{0}$ .

Facts & Assumptions

Given: The IID sampling law, finite-dimensional estimating functions, measurable estimator, and the three displayed hypotheses.

[F1]

Z-estimators: A Z-estimator is an exact or $o_{P} (1)$ -approximate root of $Ψ_{n} (θ) = P_{n} ψ_{θ}$ .

[F2]

Stochastic order and convergence notation for finite-dimensional random vectors: For fixed finite $q$ , $o_{P} (1)$ means the norm exceeds every fixed positive threshold with probability tending to zero.

[F4]

Pointwise consistency in probability: Consistency at $θ_{0}$ is convergence in probability in the metric $d$ .

[A1]

The parameter space $(Θ, d)$ is a metric space, the data are IID with law $P$ , $ψ_{θ}$ is integrable, and all displayed evaluations and suprema are measurable.

[A2]

The uniform equation error $\sup_{θ} ∥ Ψ_{n} (θ) - Ψ (θ) ∥$ is $o_{P} (1)$ .

[A3]

$Ψ (θ_{0}) = 0$ , and each displayed separation bound $b_{ε}$ is positive.

[A4]

The root residual $∥ Ψ_{n} ({\hat{θ}}_{n}) ∥$ is $o_{P} (1)$ .

Proof

1.1A1A2A3A4F2chooseF1

Fix $ε > 0$ . If ${θ : d (θ, θ_{0}) \geq ε}$ is empty, the corresponding metric-error event is empty and its probability is zero for every $n$ . Otherwise $b = b_{ε}$ is finite and positive: [A3] gives positivity, and any point in this nonempty set has a finite population equation norm by integrability in [A1]. Put $V_{n} = \sup_{θ} ∥ Ψ_{n} (θ) - Ψ (θ) ∥$ and $R_{n} = ∥ Ψ_{n} ({\hat{θ}}_{n}) ∥$ . Both are measurable, and $V_{n} = o_{P} (1)$ and $R_{n} = o_{P} (1)$ by [A2] and [A4]. The remaining steps treat this nonempty case.

2.1A1A3step 1.1algebra

On the event $d ({\hat{θ}}_{n}, θ_{0}) \geq ε$ , separation and the triangle inequality give $b \leq ∥ Ψ ({\hat{θ}}_{n}) ∥ \leq ∥ Ψ ({\hat{θ}}_{n}) - Ψ_{n} ({\hat{θ}}_{n}) ∥ + ∥ Ψ_{n} ({\hat{θ}}_{n}) ∥ \leq V_{n} + R_{n} .$ Thus the metric-error event is contained in ${V_{n} + R_{n} \geq b}$ . This uses the equation error itself, with no representation of the nonlinear quantity $- ∥ P_{n} ψ_{θ} ∥$ as an empirical average of a fixed criterion.

3.1F2F4step 1.1step 2.1given∎

Since $b > 0$ , the union bound gives $P^{\otimes n} (V_{n} + R_{n} \geq b) \leq P^{\otimes n} (V_{n} > b / 3) + P^{\otimes n} (R_{n} > b / 3) \to 0$ by [F2] and step 1.1. Step 2.1 therefore implies $P^{\otimes n} {d ({\hat{θ}}_{n}, θ_{0}) \geq ε} \to 0$ , and hence the same holds with strict inequality. Together with the empty case of step 1.1, this proves consistency for every $ε > 0$ by [F4].

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Sources

Peter Bartlett, Theoretical Statistics, Stat 210B Lecture 15, Theorem and proof, PDF pp. 8–10 (standard reference, not scraped)
Aad W. van der Vaart, Semiparametric Statistics, Lecture 5 §5.3, Theorem 5.10 and proof, PDF p. 56 (standard reference, not scraped)
Yen-Chi Chen, UW STAT 581, Chapter 3, lines 110–118 (standard reference, not scraped)