[Paper Review] On an extreme two-point distribution

V. I. Chebotarev, A. S. Kondrik+1 more|ArXiv.org|Oct 18, 2007

Probability and Risk Models3 citations

TL;DR

This paper refines the Berry–Esseen bound for the supremum distance between a standardized distribution function and the standard normal distribution. By analyzing an extreme two-point distribution, the authors prove that the optimal bound is $ C_{ ext{\Phi}} = 0.5409365\ldots $, achieved when the random variable takes values $ x_1 = -4.6925 $ and $ x_2 = 0.2131 $ with probabilities $ 0.0434 $ and $ 0.9566 $, respectively, demonstrating tightness of the bound.

ABSTRACT

A bound for functional $Δ(F)=\sup_{x\in\mathbb R}|F(x)-Φ(x)|$ is obtained, which is uniform for all distribution functions $F$ of random variables with zero mean-value and unity variance. Moreover, a two-point distribution is found, for which this bound is reached.

Motivation & Objective

• To refine the upper bound on the supremum distance between a standardized distribution function and the standard normal distribution.
• To identify the distribution that achieves the tightest possible bound in the Berry–Esseen inequality for a single random variable.
• To improve upon the previously reported bound of 0.5416 by providing a sharper, analytically derived value.
• To demonstrate that the new bound is unimprovable by constructing a specific two-point distribution that attains equality.

Proposed method

• Define the function $ \Psi(x) = \frac{1}{1+x^2} - \Phi(-|x|) $, which bounds the difference between the standard normal tail and the tail of any standardized distribution.
• Prove that $ \Psi(x) $ is positive and attains its maximum $ C_{\Phi} = 0.5409365\ldots $ at $ x = \pm x_{\Phi} $, where $ x_{\Phi} \approx 0.213105 $.
• Derive the critical point condition $ x e^{x^2/2} (1+x^2)^{-2} = (8\pi)^{-1/2} $, which defines $ x_{\Phi} $.
• Construct the extreme two-point distribution $ X_{\Phi} $ with values $ x_1 = -1/x_{\Phi} $ and $ x_2 = x_{\Phi} $, and probabilities $ p_1 = x_{\Phi}^2/(1+x_{\Phi}^2) $, $ p_2 = 1/(1+x_{\Phi}^2) $.
• Show that for this distribution, $ \Delta(F) = \sup_x |F(x) - \Phi(x)| = C_{\Phi} $, proving the bound is sharp.
• Use the inequality $ \mathbb{P}(X \geq x) \leq \frac{1}{1+x^2} $ for standardized random variables to derive the upper bound $ \Delta(F) \leq \Psi(x) \leq C_{\Phi} $.

Experimental results

Research questions

• RQ1 What is the tightest possible upper bound on the Kolmogorov-Smirnov distance between a standardized distribution and the standard normal distribution?
• RQ2 Can the previously reported bound of 0.5416 be improved, and if so, by how much?
• RQ3 Is there a specific distribution that achieves equality in the Berry–Esseen bound for a single random variable?
• RQ4 What are the exact values of the support points and probabilities of the extreme two-point distribution that saturates the bound?
• RQ5 How does the new bound compare to alternative bounds that depend on higher moments, such as the third absolute moment?

Key findings

• The supremum distance $ \Delta(F) $ between any standardized distribution function $ F $ and the standard normal distribution is bounded above by $ C_{\Phi} = 0.5409365\ldots $.
• The maximum value of $ \Psi(x) = \frac{1}{1+x^2} - \Phi(-|x|) $ is achieved at $ x = \pm x_{\Phi} $, where $ x_{\Phi} \approx 0.213105 $.
• The extreme two-point distribution $ X_{\Phi} $ with values $ x_1 = -4.692518\ldots $ and $ x_2 = 0.213105\ldots $ achieves equality in the bound $ \Delta(F) = C_{\Phi} $.
• The bound $ \Delta(F) \leq C_{\Phi} $ is unimprovable, as equality is attained for both $ F_{\Phi} $ and $ \widetilde{F}_{\Phi} $.
• The new bound $ C_{\Phi} = 0.5409365\ldots $ is strictly less than the previously reported value of 0.5416, correcting an earlier estimate.
• For distributions with third absolute moment $ \beta < C_{\Phi}/C_1 \approx 1.46 $, the new bound is tighter than alternative moment-based bounds.