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Differentiability properties of the s-mean \(m_ s\) at \(s=1\). (English) Zbl 0578.62021
Summary: It is known that the s-means \(m_ s\) of a probability measure converge for \(s\downarrow 1\) to a median \(m_ 1\) called the natural median. For purposes of estimation it is important to know whether \(m_ s\) is not only continuous but also differentiable at \(s=1\). We show that the function \(s\to m_ s\) is indeed differentiable at \(s=1\) for the case of a unique as well as a non unique median. We also give an explicit formula for the derivative at \(s=1\).
62E99 Statistical distribution theory
62B99 Sufficiency and information
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