{"ID":2837220,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2511.20905","arxiv_id":"2511.20905","title":"Nonparametric Regression for Random Unbiased Perturbations","abstract":"We study nonparametric regression with covariates $X$ and outcome $Y$ under random unbiased perturbations (RUPs) of the conditional distribution $Y|X$, where the marginal distribution of covariates, $P^X$, remains fixed but the conditional law, $P^{Y|X}$, varies randomly across datasets. Unlike adversarial distribution shift frameworks that yield conservative worst-case guarantees, RUPs induce dataset-level variance inflation rather than systematic bias. We provide examples of RUPs and show that this distributional uncertainty reduces the effective sample size to $n_{\\mathrm{eff}} = n/(1 + n τ)$, where $τ\\in [0,1]$ quantifies the perturbation strength. For local polynomial estimators, we derive an extended bias-variance decomposition that includes a distributional variance term with the same bandwidth scaling as classical sampling variance. This leads to a modified bandwidth selection principle: when distributional uncertainty dominates sampling uncertainty ($τ\\gg 1/n$), optimal bandwidths scale as $τ^{1/(2β+1)}$ rather than the usual $n^{-1/(2β+1)}$, where $β$ indicates the smoothness of the function class considered. We also establish matching minimax lower bounds showing that there exists an RUP for which this effective sample size $n_{\\mathrm{eff}}$ is fundamental. Our results demonstrate that random dataset-level perturbations create a distinct mode of uncertainty that affects both practical tuning and fundamental statistical limits.","short_abstract":"We study nonparametric regression with covariates $X$ and outcome $Y$ under random unbiased perturbations (RUPs) of the conditional distribution $Y|X$, where the marginal distribution of covariates, $P^X$, remains fixed but the conditional law, $P^{Y|X}$, varies randomly across datasets. Unlike adversarial distribution...","url_abs":"https://arxiv.org/abs/2511.20905","url_pdf":"https://arxiv.org/pdf/2511.20905v1","authors":"[\"Anna Lyubarskaja\",\"Dominik Rothenhäusler\"]","published":"2025-11-25T22:45:15Z","proceeding":"math.ST","tasks":"[\"math.ST\"]","methods":"[]","has_code":false}