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Variability of the MAX and MIN Statistic: A Theory of the Quantile Spread as a Function of Sample Size

Published online by Cambridge University Press:  01 January 2025

James T. Townsend  and
Hans Colonius
James T. Townsend*
Affiliation:
Indiana University
Hans Colonius
Affiliation:
Universitaet Oldenburg
*
Requests for reprints should be sent to James T. Townsend, Department of Psychology, Indiana University, 1101 E. 10th St. Bloomington, IN 47405-7007. E-mail: [email protected]
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Abstract

The maximum and minimum of a sample from a probability distribution are extremely important random variables in many areas of psychological theory, methodology, and statistics. For instance, the behavior of the mean of the maximum or minimum processing time, as a function of the number of component random processing times (n), has been studied extensively in an effort to identify the underlying processing architecture (e.g., Townsend & Ashby, 1983; Colonius & Vorberg, 1994). Little is known concerning how measures of variability of the maximum or minimum change with n. Here, a new measure of random variability, the quantile spread, is introduced, which possesses sufficient strength to define distributional orderings and derive a number of results concerning variability of the maximum and the minimum statistics. The quantile spread ordering may be useful in many venues. Several interesting open problems are pointed out.

Keywords

Extremal distributionsExtremal variabilityQuantile spreadParallel processing
Type
Theory and Methods
Information
Psychometrika , Volume 70 , Issue 4 , December 2005 , pp. 759 - 772
Copyright
Copyright © 2005 The Psychometric Society

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Footnotes

This work was supported by an NIH Grant R01 MH57717 to the first author. Some of the collaboration took place during the year 2000 while J.T. Townsend was a Fellow at the Hanse Institute for Advanced Study (HWK), sponsored by H. Colonius at Oldenburg University.

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