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.Eaton, Morris L. (1974) "A probability inequality for linear combinations of bounded random variables." Annals of Statistics 2(3) 609–614
Statement of the inequality
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
Eaton showed that
where φ(x) is the probability density function of the standard normal distribution.
A related bound is Edelman's
where Φ(x) is cumulative distribution function of the standard normal distribution.
Pinelis has shown that Eaton's bound can be sharpened:Pinelis, I. (1994) "Extremal probabilistic problems and Hotelling's T2 test under a symmetry condition." Annals of Statistics 22(1), 357–368
A set of critical values for Eaton's bound have been determined.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
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
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 WhittleWhittle P (1960) Bounds for the moments of linear and quadratic forms in independent variables. Teor Verojatnost i Primenen 5: 331–335 MR0133849 and p ≥ 2 was proved by Haagerup.Haagerup U (1982) The best constants in the Khinchine inequality. Studia Math 70: 231–283 MR0654838
If f(x) = eλx with λ ≥ 0 then
where inf is the infimum.Hoeffding W (1963) Probability inequalities for sums of bounded random variables. J Amer Statist Assoc 58: 13–30 MR144363
Let
ThenPinelis I (1994) Optimum bounds for the distributions of martingales in Banach spaces. Ann Probab 22(4):1679–1706
The constant in the last inequality is approximately 4.4634.
An alternative bound is also known:de la Pena, VH, Lai TL, Shao Q (2009) Self normalized processes. Springer-Verlag, New York
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 thatvan Zuijlen Martien CA (2011) On a conjecture concerning the sum of independent Rademacher random variables. https://arxiv.org/abs/1112.4988
where μ is the mean and σ is the standard deviation of the sum.
References