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September, 1954 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, K. V. Ramachandran
Ann. Math. Statist. 25(3): 565-572 (September, 1954). DOI: 10.1214/aoms/1177728724

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 $x$.

Citation

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K. C. S. Pillai. K. V. Ramachandran. "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 (3) 565 - 572, September, 1954. https://doi.org/10.1214/aoms/1177728724

Information

Published: September, 1954
First available in Project Euclid: 28 April 2007

zbMATH: 0056.37701
MathSciNet: MR64356
Digital Object Identifier: 10.1214/aoms/1177728724

Rights: Copyright © 1954 Institute of Mathematical Statistics

Vol.25 • No. 3 • September, 1954
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