## The Annals of Mathematical Statistics

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

### The Approximate Distribution of Student's Statistic

#### 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