## Accelerated convergence for nonparametric regression with coarsened predictors.(English)Zbl 1129.62032

Summary: 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 $$(W_{1}, Y_{1}), \dots , (W_n, Y_n)$$ 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, $$X=W+\delta$$. 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.

### MSC:

 62G08 Nonparametric regression and quantile regression 62G20 Asymptotic properties of nonparametric inference 60F17 Functional limit theorems; invariance principles

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### References:

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