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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

Let Xi be a set of real independent random variables, each with a expected value of zero and bounded 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

i=1nai2=1.

Eaton showed that

P(|i=1naiXi|k)2inf0ckc(zckc)3ϕ(z)dz=2BE(k),

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

A related bound is Edelman'sTemplate:Cn

P(|i=1naiXi|k)2(1Φ[k1.5k])=2BEd(k),

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

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

BEP=min{1,k2,2BE}

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

Related inequalities

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

i=1nbi2=1.

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

aibi++anbn

will converge if and only if

i=1nbi2

is finite.

Then

Ef(aibi++anbn)Ef(Z)

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

Ef(aibi++anbn)inf[E(eλZ)eλx]=ex2/2

where inf is the infimum.[6]


Let

Sn=aibi++anbn


Then[7]

P(Snx)2e39P(Zx)

The constant in the last inequality is approximately 4.4634.


An alternative bound is also known:[8]

P(Snx)ex2/2

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]

P(|μσ|)0.5Template: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. http://arxiv.org/abs/1112.4988