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On an Inequality of Hoeffding and Rosén

Jan Lanke
Scandinavian Journal of Statistics
Vol. 1, No. 2 (1974), pp. 84-86
Stable URL: http://www.jstor.org/stable/4615556
Page Count: 3
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On an Inequality of Hoeffding and Rosén
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Abstract

If X̄ and Ȳ denote the averages in two equally large samples obtained through simple random sampling with and without replacement, respectively, Hoeffding has proved that $\text{E}[\text{g}(\overline{\text{Y}}]\leq \text{E}[\text{g}(\overline{\text{X}}]$ for any convex function g while Rosén has shown that the same inequality holds also in the more general case when the X-sample has been drawn by any symmetric sampling design. Letting Z̄ denote the average of the values of the distinct units in the X-sample, we will show that $\text{E}[\text{g}(\overline{\text{Y}}]\leq \text{E}[\text{g}(\overline{\text{Z}}]\leq \text{E}[\text{g}(\overline{\text{X}}]$ holds under Rosén's assumptions.

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