Nonparametric relative error regression for spatial random variables (Q1685285)

Let \(Z=(X,Y)\) be a \(\mathbb{R}^d\times\mathbb{R}\)-valued random vector with \(Y>0\). The mean squared relative error \(E((Y-t)^2/Y^2\mid X=x)\) with respect to \(t\) is minimized by \(\vartheta(x)= E(Y^{-1}\mid X=x)/E(Y^{-2}\mid X=x)\), provided the integrals exist. Let \(Z_i\), \(i\in\mathcal{I}_n\subset\mathbb{Z}^N\) be a strictly stationary random field, consisting of copies of \(Z\). A natural estimate of \(\vartheta(x)\) is the relative error regression kernel estimate \(\tilde\vartheta_n(x)= \sum_{i\in\mathcal{I}_n}Y_i^{-1}K(h^{-1}(x-X_i))/\sum_{i\in\mathcal{I}_n}Y_i^{-2}K(h^{-1}(x-X_i))\), where \(K\) is a kernel and \(h\) a bandwidth. Under suitable regularity conditions, the authors establish uniform convergence \(\sup_{x\in S}|\tilde\vartheta_n(x)-\vartheta(x)|\to 0\) a.s. over a compact set \(S\) as well as pointwise asymptotic normality of \(\tilde\vartheta_n(x)-\vartheta(x)\), properly standardized. Confidence intervals of \(\vartheta(x)\) are derived from the asymptotic normality. A finite sample comparison with the classical kernel regression estimator is in favor of the relative error regression estimate when there are outliers in the data. An application to real data completes the paper.