Accelerated convergence for nonparametric regression with coarsened predictors
| dc.creator | Delaigle, Aurore | |
| dc.creator | Hall, Peter | |
| dc.creator | Müller, Hans-Georg | |
| dc.date | 2008-03-20 | |
| dc.date.accessioned | 2026-07-07T12:17:47Z | |
| dc.date.available | 2026-07-07T12:17:47Z | |
| dc.description | We consider nonparametric estimation of a regression function for a situation where precisely measured predictors are used to estimate the regression curve for coarsened, that is, less precise or contaminated predictors. Specifically, while one has available a sample $(W_1,Y_1),...,(W_n,Y_n)$ of independent and identically distributed data, representing observations with precisely measured predictors, where $\mathrm{E}(Y_i|W_i)=g(W_i)$, instead of the smooth regression function $g$, the target of interest is another smooth regression function $m$ that pertains to predictors $X_i$ that are noisy versions of the $W_i$. Our target is then the regression function $m(x)=E(Y|X=x)$, where $X$ is a contaminated version of $W$, that is, $X=W+δ$. It is assumed that either the density of the errors is known, or replicated data are available resembling, but not necessarily the same as, the variables $X$. In either case, and under suitable conditions, we obtain $\sqrt{n}$-rates of convergence of the proposed estimator and its derivatives, and establish a functional limit theorem. Weak convergence to a Gaussian limit process implies pointwise and uniform confidence intervals and $\sqrt{n}$-consistent estimators of extrema and zeros of $m$. It is shown that these results are preserved under more general models in which $X$ is determined by an explanatory variable. Finite sample performance is investigated in simulations and illustrated by a real data example. | |
| dc.description | Published in at http://dx.doi.org/10.1214/009053607000000497 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org) | |
| dc.identifier | https://arxiv.org/abs/0803.3017 | |
| dc.identifier | http://arxiv.org/abs/0803.3017 | |
| dc.identifier | Annals of Statistics 2007, Vol. 35, No. 6, 2639-2653 | |
| dc.identifier | doi:10.1214/009053607000000497 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/212195 | |
| dc.subject | Statistics Theory | |
| dc.subject | 62G08, 62G05 (Primary) | |
| dc.title | Accelerated convergence for nonparametric regression with coarsened predictors | |
| dc.type | text |