Abstract
It is known that thes-meanm s of a probability measure converge fors ⇃ 1 to a medianm 1 called the natural median. For purposes of estimation it is important to know whetherm s is not only continuous but also differentiable ats=1. We show that the functions→m s is indeed differentiable ats=1 for the case of a unique as well as a non unique median. We also give an explicit formula for the derivative ats=1.
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References
Jackson, D.: Note on the Median of a Set of Numbers. Bull Amer. Math. Soc.27, 1921, 160–164.
Kellerer, H.: Lageparametern-dimensionaler Verteilungen. Math. Zeitschrift73, 1960, 197–218.
Landers, D., andL. Rogge: Isotonic approximation inL s. Journal of Approximation Theory31, 1981 a, 199–223.
—: The Natural Median. Annals of Probability9, 1981 b, 1041–1042.
—: Natural choice ofL 1-approximants. Journal of Approximation Theory33, 1981 c, 268–280.
—: Consistent estimation of the Natural Median. Statistics and Decision1, 1983, 269–284.
Landers: Natural α-Quantiles and conditional α-Quantiles. To appear in Metrika31, 1984.
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Landers, D., Rogge, L. Differentiability properties of thes-Meanm s ats=1. Metrika 31, 379–384 (1984). https://doi.org/10.1007/BF01915225
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DOI: https://doi.org/10.1007/BF01915225