We study the problem of deconvolving an unknown distribution function when the error distribution is ordinary smooth and unknown. Using auxiliary data from an additional experiment that provides information about the error distribution, we establish minimax-optimal convergence rates (up to logarithmic factors), relative to the $1$-Wasserstein metric, for a kernel-based distribution function estimator over H\"{o}lder continuous spaces of densities. The analysis yields matching upper and lower bounds, ensuring the theoretical validity of the estimator. Furthermore, we propose a rate-adaptive, fully data-driven estimation procedure that automatically selects the optimal bandwidth across the $\alpha$-H\"{o}lder regularity scale of the mixing distribution with $\alpha\geq\frac{1}{2}$, requiring no prior knowledge of the oracle bandwidth.
Adaptive minimax-optimal Wasserstein deconvolution with unknown error distributions
Catia Scricciolo
2025-01-01
Abstract
We study the problem of deconvolving an unknown distribution function when the error distribution is ordinary smooth and unknown. Using auxiliary data from an additional experiment that provides information about the error distribution, we establish minimax-optimal convergence rates (up to logarithmic factors), relative to the $1$-Wasserstein metric, for a kernel-based distribution function estimator over H\"{o}lder continuous spaces of densities. The analysis yields matching upper and lower bounds, ensuring the theoretical validity of the estimator. Furthermore, we propose a rate-adaptive, fully data-driven estimation procedure that automatically selects the optimal bandwidth across the $\alpha$-H\"{o}lder regularity scale of the mixing distribution with $\alpha\geq\frac{1}{2}$, requiring no prior knowledge of the oracle bandwidth.
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