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Eaton's inequality

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In probability theory, Eaton's inequality is a bound on the largest values of a linear combination of bounded random variables. This inequality was described in 1974 by Morris L. Eaton.[1]

Statement of the inequality

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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 ≤ in). The variates do not have to be identically or symmetrically distributed. Let {ai} be a set of n fixed real numbers with

Eaton showed that

where φ(x) is the probability density function of the standard normal distribution.

A related bound is Edelman's[citation needed]

where Φ(x) is cumulative distribution function of the standard normal distribution.

Pinelis has shown that Eaton's bound can be sharpened:[2]

A set of critical values for Eaton's bound have been determined.[3]

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Let {ai} be a set of independent Rademacher random variablesP( 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

This last condition is required by the Riesz–Fischer theorem which states that

will converge if and only if

is finite.

Then

for f(x) = | x |p. The case for p ≥ 3 was proved by Whittle[4] and p ≥ 2 was proved by Haagerup.[5]


If f(x) = eλx with λ ≥ 0 then

where inf is the infimum.[6]


Let


Then[7]

The constant in the last inequality is approximately 4.4634.


An alternative bound is also known:[8]

This last bound is related to the Hoeffding's inequality.


In the uniform case where all the bi = n−1/2 the maximum value of Sn is n1/2. In this case van Zuijlen has shown that[9]

[clarification needed]

where μ is the mean and σ is the standard deviation of the sum.

References

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  1. ^ Eaton, Morris L. (1974) "A probability inequality for linear combinations of bounded random variables." Annals of Statistics 2(3) 609–614
  2. ^ Pinelis, I. (1994) "Extremal probabilistic problems and Hotelling's T2 test under a symmetry condition." Annals of Statistics 22(1), 357–368
  3. ^ 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
  4. ^ Whittle P (1960) Bounds for the moments of linear and quadratic forms in independent variables. Teor Verojatnost i Primenen 5: 331–335 MR0133849
  5. ^ Haagerup U (1982) The best constants in the Khinchine inequality. Studia Math 70: 231–283 MR0654838
  6. ^ Hoeffding W (1963) Probability inequalities for sums of bounded random variables. J Amer Statist Assoc 58: 13–30 MR144363
  7. ^ Pinelis I (1994) Optimum bounds for the distributions of martingales in Banach spaces. Ann Probab 22(4):1679–1706
  8. ^ de la Pena, VH, Lai TL, Shao Q (2009) Self normalized processes. Springer-Verlag, New York
  9. ^ van Zuijlen Martien CA (2011) On a conjecture concerning the sum of independent Rademacher random variables. https://arxiv.org/abs/1112.4988