Abstract
Let X = {X n } n≥1 and Y = {Y n } n≥1 be two independent random sequences. We obtain rates of convergence to the normal law of randomly weighted self-normalized sums
. These rates are seen to hold for the convergence of a number of important statistics, such as for instance Student’s t-statistic or the empirical correlation coefficient.
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References
H. Albrecher and J. Teugels, Asymptotic analysis of a measure of variation, Theory Probab. Math. Statist., 74 (2006), 1–9.
V. Bentkus, M. Bloznelis and F. Götze, A Berry-Esséen bound for Student’s statistic in the non-i.i.d. case, J. Theoret. Probab., 9 (1996), 765–796.
V. Bentkus, J. M. Bing-yi, Q. M. Shao and Z. Wang, Limiting distributions of the non-central t-statistic and their applications to the power of t-tests under non-normality, Bernoulli, 13 (2007), 346–364.
V. Bentkus and F. Götze, The Berry-Esséen bound for Student’s statistic, Ann. Probab., 24 (1996), 491–503.
L. Breiman, On some limit theorems similar to the arc-sin law, Teor. Veroyatn. Primen., 10 (1965), 351–359.
L. H. Y. Chen, L. Goldstein and Q-M. Shao, Normal Approximation by Stein’s Method, Springer Series in Probability and its Application, Springer, 2011.
G. P. Chistyakov and F. Götze, Limit distributions of studentized means, Ann. Probab., 32 (2004), 28–77.
B. Efron, Student’s t-test under symmetry conditions, JASA, 64 (1969), 1278–1302.
A. Fuchs, A. Joffe and J. Teugels, Expectation of the Ratio of the Sum of Squares to the Square of the Sum: Exact and Asymptotic Results, Teor. Veroyatn. Primen., 46 (2001), 297–310.
E. Giné, F. Götze and D. M. Mason, When is the student t-statistic asymptotically normal?, Ann. Probab., 25 (1997), 1514–1531.
M. Hallin, Y. Swan, T. Verdebout and D. Veredas, Rank-based testing in linear models with stable errors, J. Nonparametr. Stat., 23 (2011), 305–320.
T. L. Lai, V. De La Pena and Q. M. Shao, Self-normalized Processes: Theory and Statistical Applications, Springer Series in Probability and its Applications, Springer-Verlag, New York, 2009.
B. F. Logan, C. L. Mallows, S. O. Rice and L. A. Shepp, Limit distributions of self-normalized sums, Ann. Probab., 5 (1973), 788–809.
D. Mason and J. Zinn, When does a self-normalized weighted sum converge in distribution?, Electron. Commun. Probab., 10 (2005), 70–81.
R. Serfling, Multivariate symmetry and asymmetry, Encyclopedia of Statistical Sciences, 2nd Ed. (eds.: Kotz, Balakrishnan, Read and Vidakovic ), Wiley, 2006, 5338–5345.
Q-M. Shao, An explicit Berry-Esseen bound for the Student t-statistic via Stein’s method, Stein’s Method and Applications (eds.: Barbour and Chen ), Lecture Notes Series 5, Institute for Mathematical Sciences, NUS, 2005, 143–155.
I. G. Shevtsova, An improvement of convergence rate estimates in the Lyapunov theorem, Dokl. Math., 82 (2010), 862–864.
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Communicated by István Berkes
Research supported by the Banque Nationale de Belgique and the Communauté française de Belgique — Actions de Recherche Concertées.
Research supported by a Mandat de Chargé de Recherche from the Fonds National de la Recherche Scientifique, Communauté française de Belgique.
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Hörmann, S., Swan, Y. A note on the normal approximation error for randomly weighted self-normalized sums. Period Math Hung 67, 143–154 (2013). https://doi.org/10.1007/s10998-013-4789-8
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DOI: https://doi.org/10.1007/s10998-013-4789-8