Exact bounds on the closeness between the Student and standard normal distributions

Pinelis, Iosif

doi:10.1051/ps/2014014

Pinelis, Iosif ¹

¹ Department of Mathematical Sciences, Michigan Technological University, Houghton, 49931 Michigan, USA

ESAIM: Probability and Statistics, Tome 19 (2015), pp. 24-27

Résumé

Upper bounds on the Kolmogorov distance (and, equivalently in this case, on the total variation distance) between the Student distribution with $p$ degrees of freedom ( $S D_{p}$ ) and the standard normal distribution are obtained. These bounds are in a certain sense best possible, and the corresponding relative errors are small even for moderate values of $p$ . The same bounds hold on the closeness between $S D_{p}$ and $S D_{q}$ with $q > p$ . Comparisons with known bounds are made.

Reçu le : 2013-06-13

MR Zbl

DOI : 10.1051/ps/2014014

Classification : 62E17, 60E15, 62E20, 62E15
Keywords: Student’s distribution, standard normal distribution, Kolmogorov distance, total variation distance, probability inequalities

Affiliations des auteurs :

Pinelis, Iosif ¹

¹ Department of Mathematical Sciences, Michigan Technological University, Houghton, 49931 Michigan, USA

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     title = {Exact bounds on the closeness between the {Student} and standard normal distributions},
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     year = {2015},
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     mrnumber = {3374867},
     zbl = {1332.62054},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/ps/2014014/}
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Pinelis, Iosif. Exact bounds on the closeness between the Student and standard normal distributions. ESAIM: Probability and Statistics, Tome 19 (2015), pp. 24-27. doi: 10.1051/ps/2014014

Bibliographie
Cité par

S. Bourguin and C. Tudor, Malliavin calculus and self normalized sums. Seminaire de Propabilités XLV. Lect. Notes Math. Springer (2013) 323. | MR | Zbl

T. Cacoullos, V. Papathanasiou, and S.A. Utev, Variational inequalities with examples and an application to the central limit theorem. Ann. Probab. 22 (1994) 1607–1618. | MR | Zbl

I. Pinelis, Monotone tail and moment ratio properties of Student’s family of distributions. Version 2. (2012). | arXiv

I. Pinelis, On the Berry–Esseen bound for the Student statistic. Version 2. (2012). | arXiv

I. Pinelis, Exact bounds on the closeness between the Student and standard normal distributions. Version 2. (2013). | arXiv | MR

I. Pinelis and R. Molzon, Berry-Esséen bounds for general nonlinear statistics, with applications to Pearson’s and non-central Student’s and Hotelling’s. (2009). | arXiv

R.S. Pinkham and M.B. Wilk, Tail areas of the $t$ -distribution from a Mills’-ratio-like expansion. Ann. Math. Statist. 34 (1963) 335–337. | MR | Zbl

H. Scheffé, A useful convergence theorem for probability distributions. Ann. Math. Statist. 18 (1947) 434–438. | MR | Zbl

R. Shimizu, Error bounds for asymptotic expansion of the scale mixtures of the normal distribution. Ann. Inst. Statist. Math. 39 (1987) 611–622. | MR | Zbl

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