## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 25, Number 3 (1954), 565-572.

### On the Distribution of the Ratio of the ith Observation in an Ordered Sample from a Normal Population to an Independent Estimate of the Standard Deviation

K. C. S. Pillai and K. V. Ramachandran

#### Abstract

This paper deals with the distribution of any observation, $x_i$, in an ordered sample of size $n$ from a normal population with zero mean and unit standard deviation. The distribution has been developed as a series of Gamma functions, and has been used to obtain the distribution of $q_i = (x_i/s)$, where $s$ is an independent estimate of the standard deviation with $\nu$ degrees of freedom. In a similar manner the distribution of the Studentized maximum modulus $u_n = | x_n/s |$ has been obtained and upper 5 per cent points of $q_n$ and upper and lower 5 per cent points of $u_n$ have been given. The method of obtaining the different distributions essentially depends on appropriate expansions of the normal probability integral and its powers in the intervals $- \infty$ to $x$ and 0 to $x$.

#### Article information

**Source**

Ann. Math. Statist., Volume 25, Number 3 (1954), 565-572.

**Dates**

First available in Project Euclid: 28 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177728724

**Digital Object Identifier**

doi:10.1214/aoms/1177728724

**Mathematical Reviews number (MathSciNet)**

MR64356

**Zentralblatt MATH identifier**

0056.37701

**JSTOR**

links.jstor.org

#### Citation

Pillai, K. C. S.; Ramachandran, K. V. On the Distribution of the Ratio of the ith Observation in an Ordered Sample from a Normal Population to an Independent Estimate of the Standard Deviation. Ann. Math. Statist. 25 (1954), no. 3, 565--572. doi:10.1214/aoms/1177728724. https://projecteuclid.org/euclid.aoms/1177728724