Nonparametric relative error regression for spatial random variables

Abstract

Let \(\displaystyle Z_{\mathbf {i}}=\left( X_{\mathbf {i}},\ Y_{\mathbf {i}}\right) _{\mathbf {i}\in \mathbb {N}^{N}\, N \ge 1}\), be a \( \mathbb {R}^d\times \mathbb {R}\)-valued measurable strictly stationary spatial process. We consider the problem of estimating the regression function of \(Y_{\mathbf {i}}\) given \(X_{\mathbf {i}}\). We construct an alternative kernel estimate of the regression function based on the minimization of the mean squared relative error. Under some general mixing assumptions, the almost complete consistency and the asymptotic normality of this estimator are obtained. Its finite-sample performance is compared with a standard kernel regression estimator via a Monte Carlo study and real data example.