Asymptotic properties of the remedian

Abstract

The remedian method is a robust and storage-saving procedure for computing a central summary value of a huge data set. It mimics an ordinary sample median in many ways but needs only bk storage spaces for a sample of size n = b k. In this article, we investigate the large sample properties of the remedian method. It is shown that the remedian with any b and k such that n = bk, as an estimator of the population median, is strongly consistent as n → ∞. Furthermore, if both b and k tend to ∞ as n → ∞, the convergence rate is only slightly lower than the ordinary rate n-1/2, and the distribution of the remedian, while appropriately normalized, approaches the normal distribution. Therefore, the large sample study leads to the conclusion that the remedian method with appropriate choices of b and k does not lose much efficiency compared with the ordinary median method while dramatically reducing the storage space in computation and sustaining the robustness of median to a certain extent.