Geodesic
Second-order and bootstrap approximation to Student's $t$-statistic
Teoriâ veroâtnostej i ee primeneniâ, Tome 47 (2002) no. 2, pp. 374-381

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We prove the validity of one-term Edgeworth expansion for Student's $t$-statistic under minimal conditions: the distribution of observations is nonlattice and has finite third moment. As a corollary we obtain the second-order correctness for the bootstrap of Student's t-statistic under these optimal conditions, thus extending classical result of Singh [Ann. Statist., 9 (1981), pp. 1187–1195] to the Studentized mean.
Keywords: Student test, Edgeworth expansion, bootstrap, asymptotic expansion, self-normalized sum. 
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     author = {M. Bloznelis and H. Putter},
     title = {Second-order and bootstrap approximation to {Student's} $t$-statistic},
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     year = {2002},
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     url = {http://geodesic.mathdoc.fr/item/TVP_2002_47_2_a14/}
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M. Bloznelis; H. Putter. Second-order and bootstrap approximation to Student's $t$-statistic. Teoriâ veroâtnostej i ee primeneniâ, Tome 47 (2002) no. 2, pp. 374-381. http://geodesic.mathdoc.fr/item/TVP_2002_47_2_a14/