Eaton's inequality

Let {X_i} 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 {a_i} be a set of n fixed real numbers with

: $\backslash sum\_\{\; i\; =\; 1\; \}^n\; a\_i^2\; =\; 1\; .$

Eaton showed that

: $P\backslash left(\; \backslash left|\; \backslash sum\_\{\; i\; =\; 1\; \}^n\; a\_i\; X\_i\; \backslash right|\; \backslash ge\; k\; \backslash right)\; \backslash le\; 2\; \backslash inf\_\{\; 0\; \backslash le\; c\; \backslash le\; k\; \}\; \backslash int\_c^\backslash infty\; \backslash left(\; \backslash frac\{\; z\; -\; c\; \}\{\; k\; -\; c\; \}\; \backslash right)^3\; \backslash phi(\; z\; )\; \backslash ,\; dz\; =\; 2\; B\_E(\; k\; )\; ,$

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

A related bound is Edelman's{{cn|date=April 2013}}

: $P\backslash left(\; \backslash left|\; \backslash sum\_\{\; i\; =\; 1\; \}^n\; a\_i\; X\_i\; \backslash right|\; \backslash ge\; k\; \backslash right)\; \backslash le\; 2\; \backslash left(\; 1\; -\; \backslash Phi\backslash left[\; k\; -\; \backslash frac\{\; 1.5\; \}\{\; k\; \}\; \backslash right]\; \backslash right)\; =\; 2\; B\_\{\; Ed\; \}(\; k\; )\; ,$

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 T² test under a symmetry condition." Annals of Statistics 22(1), 357–368

: $B\_\{\; EP\; \}\; =\; \backslash min\backslash \{\; 1,\; k^\{\; -2\; \},\; 2\; B\_E\; \backslash \}$

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

Let {a_i} 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 {b_i} be a set of n fixed real numbers such that

: $\backslash sum\_\{\; i\; =\; 1\; \}^n\; b\_i^2\; =\; 1\; .$

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

:$a\_i\; b\_i\; +\; \backslash cdots\; +\; a\_n\; b\_n$

will converge if and only if

: $\backslash sum\_\{\; i\; =\; 1\; \}^n\; b\_i^2$

is finite.

Then

: $E\; f(\; a\_i\; b\_i\; +\; \backslash cdots\; +\; a\_n\; b\_n\; )\; \backslash le\; E\; f(\; Z\; )$

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

:$E\; f(\; a\_i\; b\_i\; +\; \backslash cdots\; +\; a\_n\; b\_n\; )\; \backslash le\; \backslash inf\; \backslash left[\; \backslash frac\{\; E\; (\; e^\{\; \backslash lambda\; Z\; \}\; )\; \}\{\; e^\{\; \backslash lambda\; x\; \}\; \}\; \backslash right]\; =\; e^\{\; -x^2\; /\; 2\; \}$

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

:$S\_n\; =\; a\_i\; b\_i\; +\; \backslash cdots\; +\; a\_n\; b\_n$

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

:$P(\; S\_n\; \backslash ge\; x\; )\; \backslash le\; \backslash frac\{\; 2e^3\; \}\{\; 9\; \}\; P(\; Z\; \backslash ge\; x\; )$

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

:$P(\; S\_n\; \backslash ge\; x\; )\; \backslash le\; e^\{\; -x^2\; /\; 2\; \}$

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

: $P(\; |\; \backslash mu\; -\; \backslash sigma\; |\; )\; \backslash le\; 0.5\; \backslash ,${{clarification needed|date=April 2013}}

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

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Category:Probabilistic inequalities

Category:Statistical inequalities