The asymptotic-bias dichotomy for maximum likelihood estimation of structured anomalies

Let $A$ be an anomaly in an anomaly family $\mathcal{S}$, let $\mathbf{X} \sim \operatorname{ASD}_{\mathcal{S}}(A,\mu)$, and let $\widehat{A}_{\mathcal{S}}$ be the maximum likelihood estimator. Write $\breve{\mathcal{S}}(A)$ for the sets in $\mathcal{S}$ containing $A$, and let $\mu_{\textnormal{detect}}$ denote the detection threshold.

Asymptotic-bias dichotomy. The technical condition in the unbiasedness theorem should be replaceable by $\mu \geq \mu_{\textnormal{detect}}$. Under this condition, the MLE is asymptotically biased if and only if $|\breve{\mathcal{S}}(A)|$ is exponential in $n$; equivalently,

$$\lim_{n\to\infty} \textnormal{Bias}(|\widehat{A}_{\mathcal{S}}|/n)>0$$

if $|\breve{\mathcal{S}}(A)|$ is exponential in $n$, and

$$\lim_{n\to\infty} \textnormal{Bias}(|\widehat{A}_{\mathcal{S}}|/n)=0$$

otherwise.

The conjecture asserts that the two established results—zero asymptotic bias when the number of sets containing the anomaly is sub-exponential, and positive asymptotic bias for the unstructured family—describe the only possible asymptotic bias values. The proposed relaxation from the theorem's technical assumptions to the detection condition, and the general dichotomy, remain unproved.