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2020 | OriginalPaper | Chapter

1. Non-Asymptotic Bounds

Authors : Yasunori Fujikoshi, Vladimir V. Ulyanov

Published in: Non-Asymptotic Analysis of Approximations for Multivariate Statistics

Publisher: Springer Singapore

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Abstract

Most asymptotic errors in statistical inference are based on error estimates when the sample size n and the dimension p of observations are large. More precisely, such statistical statements are evaluated when n and/or p tend to infinity. On the other hand, “non-asymptotic” results are derived under the condition that n, p, and the parameters involved are fixed. In this chapter, we explain non-asymptotic error bounds, while giving the Edgeworth expansion, Berry–Essen bounds, and high-dimensional approximations for the linear discriminant function.

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Literature
1.
go back to reference Bhattacharya, R. N., & Ghosh, J. K. (1978). On the validity of the formal Edgeworth expansion. The Annals of Statistics, 6, 434–451. (corrigendum ibid., 8, 1980).MathSciNetCrossRef Bhattacharya, R. N., & Ghosh, J. K. (1978). On the validity of the formal Edgeworth expansion. The Annals of Statistics, 6, 434–451. (corrigendum ibid., 8, 1980).MathSciNetCrossRef
2.
go back to reference Dobrić, V., & Ghosh, B. K. (1996). Some analogs of the Berry-Esseen bound for first-order Chebyshev–Edgeworth expansions. Statistics & Decision, 14, 383–404.MathSciNetMATH Dobrić, V., & Ghosh, B. K. (1996). Some analogs of the Berry-Esseen bound for first-order Chebyshev–Edgeworth expansions. Statistics & Decision, 14, 383–404.MathSciNetMATH
3.
go back to reference Fujikoshi, Y. (2000). Error bounds for asymptotic approximations of the linear discriminant function when the sample size and dimensionality are large. Journal of Multivariate Analysis, 73, 1–17.MathSciNetCrossRef Fujikoshi, Y. (2000). Error bounds for asymptotic approximations of the linear discriminant function when the sample size and dimensionality are large. Journal of Multivariate Analysis, 73, 1–17.MathSciNetCrossRef
4.
go back to reference Fujikoshi, Y., Ulyanov, V. V., & Shimizu, R. (2010). Multivariate Analysis: High-Dimensional and Large-Sample Approximations. Hoboken: Wiley.MATH Fujikoshi, Y., Ulyanov, V. V., & Shimizu, R. (2010). Multivariate Analysis: High-Dimensional and Large-Sample Approximations. Hoboken: Wiley.MATH
5.
go back to reference Shevtsova, I. G. (2011). On the absolute constants in the Berry–Esseen type inequalities for identically distributed summands. arXiv: 1111.6554. Shevtsova, I. G. (2011). On the absolute constants in the Berry–Esseen type inequalities for identically distributed summands. arXiv:​ 1111.​6554.
Metadata
Title
Non-Asymptotic Bounds
Authors
Yasunori Fujikoshi
Vladimir V. Ulyanov
Copyright Year
2020
Publisher
Springer Singapore
DOI
https://doi.org/10.1007/978-981-13-2616-5_1

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