Nonparametric Regression for Random Unbiased Perturbations

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 $n_{\mathrm{eff}} = n/(1 + n \tau)$, where $\tau\in [0,1]$ 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 ($\tau \gg 1/n$), optimal bandwidths scale as $\tau^{1/(2\beta+1)}$ rather than the usual $n^{-1/(2\beta+1)}$, where $\beta$ 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.