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Asymptotic properties of the remedian. (English) Zbl 1055.62019

Summary: 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 \(n=b^k\). We investigate the large sample properties of the remedian method. It is shown that the remedian with any \(b\) and \(k\) such that \(n=b^k\), as an estimator of the population median, is strongly consistent as \(n\to\infty\). Furthermore, if both \(b\) and \(k\) tend to \(\infty\) as \(n\to\infty\), 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.

MSC:

62E20 Asymptotic distribution theory in statistics
62G35 Nonparametric robustness
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References:

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