On the absolute constants in the Berry-Esseen-type inequalities

By Irina Shevtsova. 2011

168 citations recorded by the source corpus.

By a modification of the method that was applied in (Korolev and Shevtsova, 2010), here the inequalities $\Delta_n\leq0.3328(\beta_3+0.429)/\sqrt{n}$ and $\Delta_n\leq0.33554(\beta_3+0.415)/\sqrt{n}$ are proved for the uniform distance $\Delta_n$ between the standard normal distribution function and the distribution function of the normalized sum of an arbitrary number $n\geq1$ of independent identically distributed random variables with zero mean, unit variance and finite third absolute moment $\beta_3$. The first of these two inequalities improves one that was proved in (Korolev and Shevtsova, 2010), and as well sharpens the best known upper estimate for the absolute constant $C_0$ in the classical Berry--Esseen inequality to be $C_0<0.4756$, since $0.3328(\beta_3+0.429)\leq0.3328\cdot1.429\beta_3<0.4756\beta_3$ by virtue of the condition $\beta_3\geq1$. The second of these inequalities is also a structural improvement of the classical Berry--Esseen inequality, and as well sharpens the upper estimate for $C_0$ still more to be $C_0<0.4748$.

Subfields: Applied Mathematics, Mathematical Physics

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Citations in the corpus

Byzantine-Robust Distributed Learning: Towards Optimal Statistical Rates (2018, methodology): “2 Proof of Lemma 1 We recall the Berry-Esseen Theorem [1, 4, 7] and the bounded difference inequality, which are useful in this proof.”
Double/De-Biased Machine Learning of Global and Local Parameters Using Regularized Riesz Representers (2018, background): “The current best estimate of A is 0.4748, due to Shevtsova (2011).”
Distributed Statistical Estimation and Rates of Convergence in Normal Approximation (2017, background): “A version with an explicit constant given below is due to Shevtsova (2011).”
Mean Estimation and Regression Under Heavy-Tailed Distributions: A Survey (2019, unknown): “For the second term on the right-hand side, we may use the Berry-Esseen theorem (see Shevtsova [72]) that implies that”
Optimal-order bounds on the rate of convergence to normality in the multivariate delta method (2009, unknown): “For instance, for the linear statistic W as in (2.5) with i.i.d. ξ1, . . . , ξn, results due to Shevtsova [74] and Michel [44] imply∣∣P(W z)− Φ(z)∣∣ n(0.33554(‖ξ1‖33 + 0.415‖ξ1‖32) ∧ 30.2211‖ξ1‖33|z”
Downlink Coverage Analysis for a Finite 3-D Wireless Network of Unmanned Aerial Vehicles (2017, unknown): “4748 [37].”