Characterize the statistical properties of related diffusion-based LID estimators

Characterize the statistical properties of the likelihood-based LIDL estimator and the diffusion-based LHSD estimator, including their estimation rates and finite-sample behavior.

Background

The paper studies the population-level finite-scale FLIPD functional and derives its deterministic small-noise bias and a minimax lower bound for sample-based estimation. It does not analyze the full statistical behavior of practical related LID procedures that estimate or evaluate analogous quantities through likelihoods or diffusion models.

The conclusion identifies LIDL, which can be implemented through diffusion-model likelihood evaluation, and LHSD as related estimators whose statistical guarantees have not yet been established. The unresolved issue is therefore to determine their statistical properties rather than merely to develop implementation procedures.