On the Normal Approximation to Symmetric Binomial Distributions

Hipp, Christiane; Mattner, Lutz

doi:10.1137/s0040585x97983213

Cited by 21 publications

(17 citation statements)

References 3 publications

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“…The first three moments of X i are equal to the first three moments of the standard normal distribution but the fourth moment differs, and so by Corollary 3.2 we have a n −1 bound for the quantity |Eh(W n ) − Φh| for h ∈ C 2 b (R). However, the Kolmogorov distance between W n , which is a standardised Binomial distribution, and Z is of order n −1/2 ; see Hipp and Mattner [12]. The n −1/2 rate is also optimal with respect to Wasserstein distance as can be seen as follows.…”

Section: )mentioning

confidence: 99%

Rates of Convergence in Normal Approximation Under Moment Conditions Via New Bounds on Solutions of the Stein Equation

Gaunt

2014

J Theor Probab

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New bounds for the k-th order derivatives of the solutions of the normal and multivariate normal Stein equations are obtained. Our general order bounds involve fewer derivatives of the test function than those in the existing literature. We apply these bounds and local approach couplings to obtain an order n −(p−1)/2 bound, for smooth test functions, for the distance between the distribution of a standardised sum of independent and identically distributed random variables and the standard normal distribution when the first p moments of these distributions agree. We also obtain a bound on the convergence rate of a sequence of distributions to the normal distribution when the moment sequence converges to normal moments.

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Section: )mentioning

confidence: 99%

Rates of Convergence in Normal Approximation Under Moment Conditions Via New Bounds on Solutions of the Stein Equation

Gaunt

2014

J Theor Probab

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“…In 2007 C. Hipp and L. Mattner published an analytical proof of the inequality C 02 ≤ 1 √ 2π in the symmetric case [8]. In 2009 the second and third authors of the present paper have suggested the compound method in which a refinement of C.L.T.…”

Section: Introductionmentioning

confidence: 78%

“…Some words about bound (11). By (8), to get C 02 (N ) it is enough to calculate T (n) = sup p∈(0,0.5] T n (p) for every 1 ≤ n ≤ N , and then find max 1≤n≤N T (n). The calculation of T (n) is reduced to two problems.…”

Section: On Calculationsmentioning

confidence: 99%

On a bound of the absolute constant in the Berry–Esseen inequality for i.i.d. Bernoulli random variables

Zolotukhin¹,

Нагаев²,

Chebotarev³

2018

Modern Stochastics: Theory and Applications

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It is shown that the absolute constant in the Berry-Esseen inequality for i.i.d. Bernoulli random variables is strictly less than the Esseen constant, if 1 ≤ n ≤ 500000, where n is a number of summands. This result is got both with the help of a supercomputer and an interpolation theorem, which is proved in the paper as well. In addition, applying the method developed by S. Nagaev and V. Chebotarev in 2009-2011, an upper bound is obtained for the absolute constant in the Berry-Esseen inequality in the case under consideration, which differs from the Esseen constant by no more than 0.06%. As an auxiliary result, we prove a bound in the local Moivre-Laplace theorem which has a simple and explicit form.Despite the best possible result, obtained by J. Schulz in 2016, we propose our approach to the problem of finding the absolute constant in the Berry-Esseen inequality for two-point distributions since this approach, combining analytical methods and the use of computers, could be useful in solving other mathematical problems.Keywords Optimal value of absolute constant in Berry-Esseen inequality, binomial distribution, numerical methods 2010 MSC 60F05, 65-04

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“…I addressed Hipp and Chistyakov concerning the paper of Bentkus and Hipp and the sources of the proof of estimate (9). On July 18, 2011, Hipp answered that some results of the paper under inquiry were published in his joint work with Mattner [27] in 2007. In that paper it was demonstrated that…”

Section: Introductionmentioning

confidence: 99%

“…However, the problem of estimating the accuracy of the normal approximation for the case of arbitrary symmetric distribution of random summands was not considered in [27]. Chistyakov answered that during their meeting between 1996 and 2001 Bentkus "said rather categorically that inequality (11) was true, but he did not get around to publishing its proof."…”

Section: Introductionmentioning

confidence: 99%

Moment-Type Estimates with an Improved Structure for the Accuracy of the Normal Approximation to Distributions of Sums of Independent Symmetric Random Variables

Shevtsova¹

2013

Theory Probab. Appl.

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On the Normal Approximation to Symmetric Binomial Distributions

Abstract: The optimal constant over square root of n error bound in the central limit theorem for distribution functions of sums of independent symmetric Bernoulli random variables is 1/ √ 2πn.

Cited by 21 publications

References 3 publications

Rates of Convergence in Normal Approximation Under Moment Conditions Via New Bounds on Solutions of the Stein Equation

Rates of Convergence in Normal Approximation Under Moment Conditions Via New Bounds on Solutions of the Stein Equation

On a bound of the absolute constant in the Berry–Esseen inequality for i.i.d. Bernoulli random variables

Moment-Type Estimates with an Improved Structure for the Accuracy of the Normal Approximation to Distributions of Sums of Independent Symmetric Random Variables

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