On the asymptotic performance of median smoothers in image analysis and nonparametric regression

Abstract

For d-dimensional images and regression functions the true object is estimated by median smoothing. The mean square error of the median smoother is calculated using the framework of M-estimation, and an expression for the asymptotic rate of convergence of the mean square error is given. It is shown that the median smoother performs asymptotically as well as the local mean. The optimal window size and the bandwidth of the median smoother are given in terms of the sample size and the dimension of the problem. The rate of convergence is found to decrease as the dimension increases, and its functional dependence on the dimension changes when the dimension reaches 4.