Minimax pointwise density estimation under Wasserstein contamination

What is the pointwise squared-loss minimax rate for estimating a Hölder-smooth density in p dimensions under independent W(q,r) contamination? The contamination radius is ε and the sample size is n.

Why it matters and what counts as progress

Why it matters: A matching rate identifies both the sampling error and the unavoidable transport-contamination error.

What counts as progress: Matching lower and attainable upper bounds of max{n^(−2s/(2s+p)), ε^(2s/(s+1+p/r))} under the stated smoothness and transport assumptions.

How the problem developed

3 Aug 2023 · Initial bounds [1]
Patrick Chao and Edgar Dobriban Introduced the minimax pointwise density estimation problem under Wasserstein contamination and gave initial upper and lower bounds.
21 Nov 2025 · Sharp rate [1][2]
Patrick Chao and Edgar Dobriban, with GPT-5 Pro credited for key suggestions Released a revised analysis with the optimal minimax rate for the stated pointwise density-estimation model.

AI contribution

Solving a Research Problem in Mathematical Statistics with AI Assistance

Edgar Dobriban

For independent W(q,r) contamination and the stated Hölder-smooth clean and contaminated densities in p dimensions, revised Theorem 4.1 gives the squared-loss pointwise minimax rate max{n^(−2s/(2s+p)), ε^(2s/(s+1+p/r))}. A smoothed kernel estimator attains the upper bound.

AI involvementAI-centered

ImportanceNotable

Technical verificationFormal (human)

Human understandingFull