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December, 1946 The Approximate Distribution of Student's Statistic
Kai-Lai Chung
Ann. Math. Statist. 17(4): 447-465 (December, 1946). DOI: 10.1214/aoms/1177730884


It is well known that various statistics of a large sample (of size $n$) are approximately distributed according to the normal law. The asymptotic expansion of the distribution of the statistic in a series of powers of $n^{-\frac{1}{2}}$ with a remainder term gives the accuracy of the approximation. H. Cramer [1] first obtained the asymptotic expansion of the mean, and recently P. L. Hsu [2] has obtained that of the variance of a sample. In the present paper we extend the Cramer-Hsu method to Student's statistic. The theorem proved states essentially that if the population distribution is non-singular and if the existence of a sufficient number of moments is assumed, then an asymptotic expansion can be obtained with the appropriate remainder. The first four terms of the expansion are exhibited in formula (35).


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Kai-Lai Chung. "The Approximate Distribution of Student's Statistic." Ann. Math. Statist. 17 (4) 447 - 465, December, 1946.


Published: December, 1946
First available in Project Euclid: 28 April 2007

MathSciNet: MR18390
Digital Object Identifier: 10.1214/aoms/1177730884

Rights: Copyright © 1946 Institute of Mathematical Statistics


Vol.17 • No. 4 • December, 1946
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