Asymptotic properties for median cross-validated nearest neighbor median estimate in nonparametric regression

Abstract

Consider the nonparametric regression model$Y_{ni} = g\left( {x_{ni} } \right) + e_{ni} ,1 \leqslant i \leqslant n$, whereg is an unknown function to be estimated on [0, 1],$x_{ni} \left( {1 \leqslant i \leqslant n} \right)$ are the fixed design points in the interval [0, 1] and$\{ e_{ni} ,1 \leqslant i \leqslant n\}$ is a triangular array of row iid random variables having median zero. The nearest neighbor median estimator$\hat g_{n,h} \left( {x_{ni} } \right) = \hat m\left( {Y_{i(l)}^n , \cdots ,Y_{i(h)}^n } \right)$ is taken as the estimator of the unknown functiong(x). Median cross validation (mev) criterion is employed to select the smoothing parameterh. Leth ^*_π be the smoothing parameter chosen by mev criterion. Under mild regularity conditions, the upper and lower bounds ofh ^*_π , the rate of convergence and the weak consistency of the median cross-validated estimate$\hat g_{n,h_n^ * } \left( {x_{ni} } \right)$ are obtained.