On the real accuracy of approximation in the central limit theorem

Senatov, V. V.

doi:10.1134/s003744661104015x

Cited by 7 publications

(4 citation statements)

References 6 publications

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“…In the i.i.d. case and under some smoothness conditions, Senatov (2011) obtains such improved bounds. To our knowledge, the question is nevertheless still open in the i.n.i.d.…”

Section: Introductionmentioning

confidence: 86%

Explicit non-asymptotic bounds for the distance to the first-order Edgeworth expansion

Derumigny¹,

Girard²,

Guyonvarch³

2021

Preprint

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In this article, we study bounds on the uniform distance between the cumulative distribution function of a standardized sum of independent centered random variables with moments of order four and its first-order Edgeworth expansion. Existing bounds are sharpened in two frameworks: when the variables are independent but not identically distributed and in the case of independent and identically distributed random variables. Improvements of these bounds are derived if the third moment of the distribution is zero. We also provide adapted versions of these bounds under additional regularity constraints on the tail behavior of the characteristic function. We finally present an application of our results to the lack of validity of one-sided tests based on the normal approximation of the mean for a fixed sample size.

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“…In the i.i.d. case and under some smoothness conditions, Senatov (2011) obtains such improved bounds. To our knowledge, the question is nevertheless still open in the i.n.i.d.…”

Section: Introductionmentioning

confidence: 86%

Explicit non-asymptotic bounds for the distance to the first-order Edgeworth expansion

Derumigny¹,

Girard²,

Guyonvarch³

2021

Preprint

View full text Add to dashboard Cite

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“…In contrast to similar problems related to the central limit theorem, [29], [23], [35], [36], the construction of asymptotic expansions and accompanying measures in limit theorems for maximums is much simpler, in a certain sense even trivial. Similar evaluations based on Taylor expansions one can find in many works on extreme distributions, see, for example, [27] and references given in Section 3.…”

Section: Asymptotic Expansions and Accompanying Measuresmentioning

confidence: 99%

“…Similarly to representation (34), using (35) with corresponding modification δ(t) on a finite interval, we get by simple calculus, that…”

Section: Log-weibull-like Distributionsmentioning

confidence: 99%

“…Notice that this is very common approach in the study of quality approximation in Central Limit Theorem. This is Chebyshev-Hermite polynomials approximation, other types of approximation, other types of accompanying laws and charges (signed measures), see [12], [16], [23], [29], [36], [35]. In connection with, in 2002, one of the authors discussed with Laurens de Haan the following result on Gaussian smooth stationary processes, only recently it is published in [30], we then have agreed that such approach is interesting and promising.…”

Section: Introductionmentioning

confidence: 99%

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On accompanying measures and asymptotic expansions in limit theorems for maximum of random variables

Piterbarg¹,

Scherbakova²

2020

Preprint

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A sequence of accompanying laws is suggested in the limit theorem of B. V. Gnedenko for maximums of independent random variables belonging to maximum domain of attraction of the Gumbel distribution. It is shown that this sequence gives an exponential power rate of convergence whereas the Gumbel distribution gives only a logarithmic rate. As examples, classes of Weibull and log-Weibull type distributions are considered in details. A scale for the Gumbel maximum domain of attraction is suggested as a continuation of the considered two classes.

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Explicit Non-Asymptotic Bounds for the Distance to the First-Order Edgeworth Expansion

Derumigny,

Girard,

Guyonvarch

2023

Sankhya A

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On the real accuracy of approximation in the central limit theorem

Cited by 7 publications

References 6 publications

Explicit non-asymptotic bounds for the distance to the first-order Edgeworth expansion

Explicit non-asymptotic bounds for the distance to the first-order Edgeworth expansion

On accompanying measures and asymptotic expansions in limit theorems for maximum of random variables

Explicit Non-Asymptotic Bounds for the Distance to the First-Order Edgeworth Expansion

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