An upper bound for the absolute constant in the nonuniform version of the Berry-Esseen inequalities for nonidentically distributed summands

Григорьева, М.Е.; Popov, S. V.

doi:10.1134/s1064562412040242

Cited by 4 publications

(4 citation statements)

References 4 publications

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Section: A Historical Sketch Of the Problem Of Nonuniform Be Boundsmentioning

confidence: 92%

“…Finally, working along lines quite similar to those in [23], Grigor'eva and Popov [11,10] claimed that c nu ă 22.2417 in the general, non-iid case. However, there appears to be the same kind of errors there: compare [10, (9) and (11)] with [23, (14) and ( 16)], respectively. This leaves, for now, 31.935 as the best (possibly correctly) established nonuniform BE constant factor c nu -in the general, non-iid case.…”

Section: A Historical Sketch Of the Problem Of Nonuniform Be Boundsmentioning

confidence: 92%

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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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Section: A Historical Sketch Of the Problem Of Nonuniform Be Boundsmentioning

confidence: 92%

Section: A Historical Sketch Of the Problem Of Nonuniform Be Boundsmentioning

confidence: 92%

On the Nonuniform Berry–Esseen Bound

Pinelis

2017

Inequalities and Extremal Problems in Probability and Statistics

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“…Remark 2. Condition (3) in Theorem 1 is used to bound the residual term in normal approximation by the nonuniform Berry-Esseen Theorem (Grigor'eva and Popov, 2012). Since the minimal of the left hand side is n −1/2 (corresponding to a WNN classifier where every data point has an equal vote of 1/n,) this condition suggests that n −1/2 = o(s −1/2 (log(s)) −2 ), i.e., roughly speaking, s/n = o(1), or γ < 1/2.…”

Section: Dnn Classifier Via Majority Votingmentioning

confidence: 99%

Distributed Nearest Neighbor Classification

Duan¹,

Qiao²,

Cheng³

2018

Preprint

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Nearest neighbor is a popular nonparametric method for classification and regression with many appealing properties. In the big data era, the sheer volume and spatial/temporal disparity of big data may prohibit centrally processing and storing the data. This has imposed considerable hurdle for nearest neighbor predictions since the entire training data must be memorized. One effective way to overcome this issue is the distributed learning framework. Through majority voting, the distributed nearest neighbor classifier achieves the same rate of convergence as its oracle version in terms of both the regret and instability, up to a multiplicative constant that depends solely on the data dimension. The multiplicative difference can be eliminated by replacing majority voting with the weighted voting scheme. In addition, we provide sharp theoretical upper bounds of the number of subsamples in order for the distributed nearest neighbor classifier to reach the optimal convergence rate. It is interesting to note that the weighted voting scheme allows a larger number of subsamples than the majority voting one. Our findings are supported by numerical studies using both simulated and real data sets.

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“…and in its non-uniform analogues for sums of independent r.v. 's that use the Berry-Esseen inequality with an improved structure (see [2,18,8]), as well as in the moment-type estimates of the rate of convergence in limit theorems for compound and mixed compound Poisson distributions (where β 3 → ∞, see [13,10,17]) which use the Berry-Esseen inequality with an improved structure as well.…”

Section: Motivation and Applicationsmentioning

confidence: 99%

A square bias transformation of probability distributions: Some properties and applications

Shevtsova

2013

Dokl. Math.

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The properties of the square bias transformation are studied, in particular, the precise moment-type estimate for the L 1 -metric between the transformed and the original distributions is proved, a relation between their characteristic functions is found. As a corollary, some new moment-type estimates for the proximity of arbitrary characteristic function with zero mean and finite third moment to the normal one with zero mean and the same variance are proved involving the double integrals of the square-and zero-bias transformations.

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An upper bound for the absolute constant in the nonuniform version of the Berry-Esseen inequalities for nonidentically distributed summands

Cited by 4 publications

References 4 publications

On the Nonuniform Berry–Esseen Bound

On the Nonuniform Berry–Esseen Bound

Distributed Nearest Neighbor Classification

A square bias transformation of probability distributions: Some properties and applications

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