Eaton's Inequality

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 be a set of real independent random variables, each with an expected value of zero and bounded above by 1 ( , ''X''_''i'' , ≤ 1, for 1 ≤ ''i'' ≤ ''n''). The variates do not have to be identically or symmetrically distributed. Let be a set of ''n'' fixed real numbers with

: $\sum_^n a_i^2 = 1 .$

Eaton showed that

: $P\left( \left, \sum_^n a_i X_i \ \ge k \right) \le 2 \inf_ \int_c^\infty \left( \frac \right)^3 \phi( z ) \, dz = 2 B_E( k ) ,$

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

A related bound is Edelman's

: P\left( \left, \sum_^n a_i X_i \ \ge k \right) \le 2 \left( 1 - \Phi\left k - \frac \right\right) = 2 B_( k ) ,

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 ''T''² test under a symmetry condition." ''Annals of Statistics'' 22(1), 357–368

: $B_ = \min\$

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 be a set of independent Rademacher random variables – ''P''( ''a''_''i'' = 1 ) = ''P''( ''a''_''i'' = −1 ) = 1/2. Let ''Z'' be a normally distributed variate with a mean 0 and variance of 1. Let be a set of ''n'' fixed real numbers such that

: $\sum_^n b_i^2 = 1 .$

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

:$a_i b_i + \cdots + a_n b_n$

will converge if and only if

: $\sum_^n b_i^2$

is finite.

Then

: $E f( a_i b_i + \cdots + a_n b_n ) \le E f( Z )$

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) = ''e''^λx with ''λ'' ≥ 0 then

: E f( a_i b_i + \cdots + a_n b_n ) \le \inf \left \frac \right= e^

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

:$S_n = a_i b_i + \cdots + a_n b_n$

ThenPinelis I (1994) Optimum bounds for the distributions of martingales in Banach spaces. Ann Probab 22(4):1679–1706

:$P( S_n \ge x ) \le \frac P( Z \ge x )$

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

:$P( S_n \ge x ) \le e^$

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

In the uniform case where all the ''b''_''i'' = ''n''^−1/2 the maximum value of ''S''_''n'' is ''n''^1/2. 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

: $P( , \mu - \sigma , ) \le 0.5 \,$

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

References

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Probabilistic inequalities
Statistical inequalities