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The Pitman inequality for exchangeable random vectors. (English) Zbl 1287.60026

Summary: In this short article the following inequality called the ‘Pitman inequality’ is proved for the exchangeable random vector \((X_1,X_2,\dots,X_n)\) without the assumption of continuity and symmetry for each component \(X_i\): \[ \operatorname P\left(\Bigg|\frac{1}{n}\sum^n_{i=1}X_i\Bigg|\leq\Bigg|\sum^n_{i=1}\alpha_iX_i\Bigg|\right)\geq\frac{1}{2}, \] where all \(\alpha_i\geq0\) are special weights with \(\sum^n_{i=1}\alpha_i=1\).

MSC:

60E15 Inequalities; stochastic orderings
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References:

[1] Balakrishnan, N.; Iliopoulos, G.; Keating, J. P.; Mason, R. L., Pitman closeness of sample median to population median, Statistics and Probability Letters, 79, 1759-1766 (2009) · Zbl 1169.62324
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[3] Chow, Y. S.; Teicher, H., Probability Theory, Independence and Interchangeability (1997), Springer · Zbl 0891.60002
[4] Ghosh, M.; Sen, P. K., Median unbiasedness and Pitman-closeness, Journal of the American Statistical Association, 84, 1089-1091 (1989) · Zbl 0702.62022
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