The Annals of Mathematical Statistics

The Approximate Distribution of Student's Statistic

Kai-Lai Chung

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Abstract

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).

Article information

Source
Ann. Math. Statist. Volume 17, Number 4 (1946), 447-465.

Dates
First available in Project Euclid: 28 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177730884

Digital Object Identifier
doi:10.1214/aoms/1177730884

Mathematical Reviews number (MathSciNet)
MR18390

JSTOR
links.jstor.org

Citation

Chung, Kai-Lai. The Approximate Distribution of Student's Statistic. Ann. Math. Statist. 17 (1946), no. 4, 447--465. doi:10.1214/aoms/1177730884. https://projecteuclid.org/euclid.aoms/1177730884


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