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An exact M-estimator can fail without population separation

Statement refuted

Without a separated population maximizer, every exact empirical M-estimator is still consistent for any fixed population maximizer.

Facts & Assumptions

Given: The sample space $X = {0}$ with its full power-set sigma-algebra and IID observations with $P (X = 0) = 1$ ; the parameter space is $Θ = {0, 1}$ with distance $d (θ, η) = ∣ θ - η ∣$ , and $m_{θ} (x) = 0$ for both parameters.

[F1]

M-estimators: An exact M-estimator is a measurable estimator that attains the empirical criterion maximum.

[F2]

Pointwise consistency in probability: Consistency for a fixed target requires the probability of each fixed positive metric error to tend to zero.

[F3]

The absolute value makes $R$ a metric space: $d (x, y) = ∣ x - y ∣$ is a metric, its open balls are the intervals $(x - r, x + r)$ , and it is unbounded: The usual metric on $R$ is $d (x, y) = ∣ x - y ∣$ ; on $Θ = {0, 1}$ the distance used here is its restriction, which remains a metric because the metric axioms persist on subsets.

[F4]

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

[A1]

The observations are IID with common law $δ_{0}$ , and the fixed population maximizer under consideration is $θ_{0} = 0$ .

Counterexample

Given: The objects and hypotheses specified above.

1.1A1F1F4given

By [A1] and [F4], every observation equals $0$ almost surely and the sequence is IID. Thus $P_{n} m_{0} = P_{n} m_{1} = P m_{0} = P m_{1} = 0$ for every $n$ . Both parameters are population maximizers, and every data-measurable choice is an exact M-estimator by [F1]. The uniform error is identically zero, but the population maximum is not separated from the other parameter.

2.1A1F1F2F3givenstep 1.1

Choose the deterministic estimator ${\hat{θ}}_{n} = 1$ for odd $n$ and ${\hat{θ}}_{n} = 0$ for even $n$ . For each $n$ it is a constant function of the data, so every inverse image is empty or the full sample space and it is measurable; by step 1.1 it is exact. For the fixed population maximizer $θ_{0} = 0$ and $ε = 1 / 2$ , $P {d ({\hat{θ}}_{n}, θ_{0}) > ε}$ equals $1$ for odd $n$ and $0$ for even $n$ , so the error probability has no zero limit.

3.1F2step 2.1given∎

This exact M-estimator is not consistent for $θ_{0} = 0$ by [F2]. Thus the unseparated population maximum does not guarantee consistency, even when the empirical criterion equals its population version exactly.

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Sources

Peter Bartlett, Theoretical Statistics, Stat 210B Lecture 15, separation hypothesis and proof, PDF pp. 6–8 (standard reference, not scraped)