On estimates of the remainder term in the central limit theorem

Bikelis, A.

doi:10.15388/lmj.1966.19732

Cited by 23 publications

(24 citation statements)

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“…for all real z ě 0, where c nu is an absolute constant. Bikelis [2] extended this result to the case of non-iid X i 's. Nagaev's method involves the following essential components:…”

Section: Nonuniform Be Bounds: Nagaev's Results and Methodsmentioning

confidence: 99%

“…The value of the truncation level y is chosen (i) to be large enough so that the tails of the truncated sum S pyq :" X pyq 1 `¨¨¨`X pyq n be close enough to those of S and, on the other hand, (ii) to be small enough so that the exponential tilt and the exponential inequality result in not too large a bound. In some variants, including the ones in [21,2], two different truncation levels are used.…”

Section: Nonuniform Be Bounds: Nagaev's Results and Methodsmentioning

confidence: 99%

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On the Nonuniform Berry–Esseen Bound

Pinelis

2017

Inequalities and Extremal Problems in Probability and Statistics

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Certain smoothing inequalities were proposed in the recent paper posted on arXiv at http://arxiv.org/abs/1301.2828 in order to lessen the large gap between the best correctly established upper and lower bounds on the constant factor in the nonuniform Berry-Esseen bound. As an illustration of the possible uses of those inequalities, a quick proof of Nagaev's classical nonuniform bound was given there. Here we describe another, apparently more effective class of smoothing inequalities, and give a yet quicker and simpler proof of Nagaev's result.AMS 2010 subject classifications: 60E15, 62E17.

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“…for all real z ě 0, where c nu is an absolute constant. Bikelis [2] extended this result to the case of non-iid X i 's. Nagaev's method involves the following essential components:…”

Section: Nonuniform Be Bounds: Nagaev's Results and Methodsmentioning

confidence: 99%

Section: Nonuniform Be Bounds: Nagaev's Results and Methodsmentioning

confidence: 99%

On the Nonuniform Berry–Esseen Bound

Pinelis

2017

Inequalities and Extremal Problems in Probability and Statistics

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“…By "universal constant" we mean that the constant is independent of P X 1 . Inequality (22) has been proven by Nagaev [14] and Bikelis [4] for λ = 3 and λ ∈ (2, 3], respectively. Meanwhile there exist several estimates for the constant C λ for λ ∈ (2, 3]; see [15] and references cited therein.…”

Section: Proofsmentioning

confidence: 94%

Nonparametric estimation of risk measures of collective risks

Lauer

Zähle

2015

Statistics &Amp; Risk Modeling

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We consider two nonparametric estimators for the risk measure of the sum of n i.i.d. individual insurance risks where the number of historical single claims that are used for the statistical estimation is of order n. This framework matches the situation that nonlife insurance companies are faced with within in the scope of premium calculation. Indeed, the risk measure of the aggregate risk divided by n can be seen as a suitable premium for each of the individual risks. For both estimators divided by n we derive a sort of Marcinkiewicz-Zygmund strong law as well as a weak limit theorem. The behavior of the estimators for small to moderate n is studied by means of Monte-Carlo simulations.

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“…These were improved to CnE|X 1 | 3 /(1 + |x| 3 ) by Nagaev (1965). Bikelis (1966) generalized Nagaev's result to…”

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confidence: 89%

Normal approximation under local dependence

Chen¹,

Shao²

2004

Ann. Probab.

168

189

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We establish both uniform and nonuniform error bounds of the Berry-Esseen type in normal approximation under local dependence. These results are of an order close to the best possible if not best possible. They are more general or sharper than many existing ones in the literature. The proofs couple Stein's method with the concentration inequality approach.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000045

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On estimates of the remainder term in the central limit theorem

Cited by 23 publications

References 0 publications

On the Nonuniform Berry–Esseen Bound

On the Nonuniform Berry–Esseen Bound

Nonparametric estimation of risk measures of collective risks

Normal approximation under local dependence

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