Develop deconvolution theory with unknown error correlation

Develop multivariate cumulative distribution function deconvolution methods and convergence-rate theory when the correlation structure of the additive measurement error is unknown.

Background

The supplement explains that if the error vector is a known invertible linear transformation of a vector with independent components, the observations can be transformed to reduce the problem to the independent-error setting. The resulting rates transfer back under the known transformation.

When the error covariance or transformation matrix is unknown, this reduction cannot be directly applied. The supplement notes that estimating the transformation from replicated pure-noise data may introduce additional error, while a completely unknown transformation leads to a blind-deconvolution problem. The theoretical treatment of unknown correlation is left unresolved.

References

Extending the theoretical results to unknown correlation remains an open problem.

— Adaptive minimax multivariate \(L^1\)-deconvolution: harmonic-mean rates under noise filtering, and a \(1\)-Wasserstein equivalence  (2610.11760 - Scricciolo, 8 Oct 2026) in Supplement, Section S3, subsection “Linear transformations of errors with independent components”