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Consistency of M-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 $m_{θ}$ be a jointly measurable, $P$ -integrable real criterion for every $θ \in Θ$ . Write $M_{n} (θ) = P_{n} m_{θ}$ and $M (θ) = P m_{θ}$ . Fix $θ_{0} \in Θ$ . Assume the displayed suprema and estimator evaluations below are measurable and finite where used. In the maximization case, suppose

$U_{n} = \sup_{θ \in Θ} ∣ M_{n} (θ) - M (θ) ∣ = o_{P} (1)$ ;
for every $ε > 0$ , $γ_{ε} = M (θ_{0}) - \sup {M (θ) : d (θ, θ_{0}) \geq ε} > 0$ , with $\sup \emptyset = - \infty$ ;
the measurable estimator ${\hat{θ}}_{n}$ satisfies $M_{n} ({\hat{θ}}_{n}) \geq M_{n} (θ_{0}) - ρ_{n}$ for a nonnegative $ρ_{n} = o_{P} (1)$ .

Then ${\hat{θ}}_{n}$ is consistent in probability for $θ_{0}$ , meaning $P^{\otimes n} {d ({\hat{θ}}_{n}, θ_{0}) > ε} \to 0$ for every $ε > 0$ (Pointwise consistency in probability). The same conclusion holds for a minimizing estimator if, for every $ε > 0$ , $\inf {M (θ) : d (θ, θ_{0}) \geq ε} - M (θ_{0}) > 0$ and $M_{n} ({\hat{θ}}_{n}) \leq M_{n} (θ_{0}) + σ_{n}$ for a nonnegative measurable $σ_{n} = o_{P} (1)$ . The relative criterion-gap premise in the maximizing case is implied by the global exact or $o_{P} (1)$ -approximate M-estimator definition, as proved in the final proof step.

The uniform-convergence premise is an explicit hypothesis. This theorem does not replace it with pointwise convergence or a pointwise law of large numbers.

Facts & Assumptions

Given: The IID sampling law, metric parameter space, measurable criterion and estimator, and the three displayed maximization hypotheses; also [A6] when invoking the global-M-estimator specialization.

[F1]

M-estimators: An M-estimator is a data-measurable exact or $o_{P} (1)$ -approximate maximizer or minimizer of the empirical criterion $P_{n} m_{θ}$ .

[F2]

Stochastic order and convergence notation for finite-dimensional random vectors: $o_{P} (1)$ for a real random variable means convergence in probability to zero; for finite-dimensional vectors it means the norm exceeds each fixed positive threshold with probability tending to zero.

[F3]

Pointwise consistency in probability: At a fixed target, consistency means that the probability of every fixed positive metric error tends to zero.

[A1]

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

[A2]

The uniform error $U_{n}$ is $o_{P} (1)$ .

[A3]

For every $ε > 0$ , the population criterion has the stated positive separation gap $γ_{ε}$ .

[A4]

The empirical maximization error $ρ_{n}$ is nonnegative and $o_{P} (1)$ , and $M_{n} ({\hat{θ}}_{n}) \geq M_{n} (θ_{0}) - ρ_{n}$ .

[A5]

For the minimizing branch, the population minimum has the stated positive separation gap and the nonnegative measurable minimization error $σ_{n}$ is $o_{P} (1)$ with $M_{n} ({\hat{θ}}_{n}) \leq M_{n} (θ_{0}) + σ_{n}$ .

[A6]

For the global M-estimator specialization, the nonnegative empirical gap $τ_{n} = \sup_{θ \in Θ} M_{n} (θ) - M_{n} ({\hat{θ}}_{n})$ is finite and $o_{P} (1)$ ; exact maximizers have $τ_{n} = 0$ .

Proof

1.1A1A3choose

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 write $γ = γ_{ε}$ , which is finite and positive because the nonempty set contains a point with finite population criterion and [A3] bounds its supremum strictly below the finite $M (θ_{0})$ . On the event $d ({\hat{θ}}_{n}, θ_{0}) \geq ε$ , separation gives $M ({\hat{θ}}_{n}) \leq M (θ_{0}) - γ$ . The remaining steps treat this nonempty case.

2.1A2A3A4step 1.1algebra

On that event, [A2] gives $M_{n} ({\hat{θ}}_{n}) \leq M (θ_{0}) - γ + U_{n}$ and $M_{n} (θ_{0}) \geq M (θ_{0}) - U_{n}$ . The approximate-maximization inequality [A4] also gives $M_{n} ({\hat{θ}}_{n}) \geq M_{n} (θ_{0}) - ρ_{n}$ . Combining them yields $γ \leq 2 U_{n} + ρ_{n}$ .

3.1F2F3A2A4step 2.1given

Since $U_{n} = o_{P} (1)$ and $ρ_{n} = o_{P} (1)$ , the probability of $2 U_{n} + ρ_{n} \geq γ$ tends to zero. Step 2.1 therefore gives $P^{\otimes n} {d ({\hat{θ}}_{n}, θ_{0}) \geq ε} \to 0$ , which implies the strict-error probability in [F3] tends to zero. Thus ${\hat{θ}}_{n}$ is consistent.

4.1A1A2A5step 1.1step 2.1step 3.1algebra

For the minimizing case, set ${\tilde{M}}_{n} = - M_{n}$ and $\tilde{M} = - M$ . By [A5], the separated-minimum condition becomes separated maximization of $\tilde{M}$ , and ${\tilde{M}}_{n} ({\hat{θ}}_{n}) \geq {\tilde{M}}_{n} (θ_{0}) - σ_{n}$ . By [A2], $\sup_{θ} ∣ {\tilde{M}}_{n} (θ) - \tilde{M} (θ) ∣ = U_{n} = o_{P} (1)$ . Thus the maximizing hypotheses in steps 1.1–3.1 hold for the transformed criterion, proving consistency in the minimizing case as well.

5.1A4A6F1algebra∎

For an exact or approximate global M-estimator, [F1] and [A6] give $M_{n} ({\hat{θ}}_{n}) \geq \sup_{θ \in Θ} M_{n} (θ) - τ_{n}$ . Since $M_{n} (θ_{0}) \leq \sup_{θ \in Θ} M_{n} (θ)$ , this implies $M_{n} ({\hat{θ}}_{n}) \geq M_{n} (θ_{0}) - τ_{n}$ , which is the relative maximization condition [A4] with $ρ_{n} = τ_{n}$ . Thus the theorem applies.

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Sources

Peter Bartlett, Theoretical Statistics, Stat 210B Lecture 15, Theorem and proof, PDF pp. 7–8 (standard reference, not scraped)
Aad W. van der Vaart, Semiparametric Statistics, Lecture 5 §5.3, Theorem 5.8 and proof, PDF p. 54 (standard reference, not scraped)
Yen-Chi Chen, UW STAT 581, Chapter 3, lines 100–109 (standard reference, not scraped)