Accelerated convergence for nonparametric regression with coarsened predictors

Abstract

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 (W₁, Y₁), …, (W_n, Y_n) of independent and identically distributed data, representing observations with precisely measured predictors, where 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.