Let {Xi} be a set of real independent random variables, each with an expected value of zero and bounded above by 1 ( |Xi | ≤ 1, for 1 ≤ i ≤ n). The variates do not have to be identically or symmetrically distributed. Let {ai} be a set of n fixed real numbers with
Pinelis has shown that Eaton's bound can be sharpened:[2]
A set of critical values for Eaton's bound have been determined.[3]
Related inequalities
Let {ai} be a set of independent Rademacher random variables – P( ai = 1 ) = P( ai = −1 ) = 1/2. Let Z be a normally distributed variate with a mean 0 and variance of 1. Let {bi} be a set of n fixed real numbers such that
↑ Eaton, Morris L. (1974) "A probability inequality for linear combinations of bounded random variables." Annals of Statistics 2(3) 609–614
↑ Pinelis, I. (1994) "Extremal probabilistic problems and Hotelling's T2 test under a symmetry condition." Annals of Statistics 22(1), 357–368
↑ Dufour, J-M; Hallin, M (1993) "Improved Eaton bounds for linear combinations of bounded random variables, with statistical applications", Journal of the American Statistical Association, 88(243) 1026–1033
↑ Whittle P (1960) Bounds for the moments of linear and quadratic forms in independent variables. Teor Verojatnost i Primenen 5: 331–335 MR0133849
↑ Haagerup U (1982) The best constants in the Khinchine inequality. Studia Math 70: 231–283 MR0654838
↑ Hoeffding W (1963) Probability inequalities for sums of bounded random variables. J Amer Statist Assoc 58: 13–30 MR144363
↑ Pinelis I (1994) Optimum bounds for the distributions of martingales in Banach spaces. Ann Probab 22(4):1679–1706
↑ de la Pena, VH, Lai TL, Shao Q (2009) Self normalized processes. Springer-Verlag, New York
↑ van Zuijlen Martien CA (2011) On a conjecture concerning the sum of independent Rademacher random variables. https://arxiv.org/abs/1112.4988
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