Hostname: page-component-745bb68f8f-hvd4g Total loading time: 0 Render date: 2025-01-08T12:05:25.288Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.

References

Arnold, B.C. (1987). Majorization and the Lorenz order: A brief introduction. New York: Springer-Verlag.CrossRefGoogle Scholar
Arnold, B.C., Groeneveld, R.A. (1995). Measuring skewness with respect to the mode. The American Statistician, 49, 34–38.CrossRefGoogle Scholar
Ashby, F.G., Townsend, J.T. (1986). Varieties of perceptual independence. Psychological Review, 93, 154–179.CrossRefGoogle ScholarPubMed
Belzunce, F. (1999). On a characterization of right spread order by the increasing convex order. Statistics & Probability Letters, 45, 103–110.CrossRefGoogle Scholar
Bernstein, I.H. (1970). Can we see and hear at the same time? Some recent studies of intersensory facilitation of reaction time. Acta Psychologica, 33, 21–35.CrossRefGoogle ScholarPubMed
Colonius, H. (1990). Possibly dependent probability summation of reaction time. Journal of Mathematical Psychology, 34, 253–275.CrossRefGoogle Scholar
Colonius, H., Vorberg, D. (1994). Distribution inequalities for parallel models with unlimited capacity. Journal of Mathematical Psychology, 38, 35–58.CrossRefGoogle Scholar
Cox, D.R. (1962). Renewal theory. London: Methuen.Google Scholar
Donnelley, N., Found, A., Müller, H.J. (1999). Searching for impossible objects: Processing form and attributes in early vision. Perception & Psychophysics, 61, 675–690.CrossRefGoogle Scholar
Dzhafarov, E.N. (1992). The structure of simple reaction time to step-function signals. Journal of Mathematical Psychology, 36, 235–268.CrossRefGoogle Scholar
Egeth, H.E. (1966). Parallel vs. serial processes in multidimensional stimulus discrimination. Perception & Psychophysics, 1, 245–252.CrossRefGoogle Scholar
Gnedenko, B.V. (1943). Sur la distribution limited u terme maximum d'une serie aleatorie. Annals of Mathematics, 44, 423–453.CrossRefGoogle Scholar
Gumbel, E.J. (1958). Statistics of extremes. New York: Columbia University Press.CrossRefGoogle Scholar
Harter, H.L. (1978). A bibliography of extreme-value theory. International Statistical Review, 46, 279–306.Google Scholar
Hartley, H.O., David, H.A. (1951). Universal bounds for mean spread and extreme observation. Annals of Mathematical Statistics, 25, 85–99.CrossRefGoogle Scholar
Hughes, H.C., Townsend, J.T. (1998). Varieties of binocular interaction in human vision. Psychological Science, 9, 53–60.CrossRefGoogle Scholar
Kleiber, C. (2002). Variability ordering of heavy-tailed distributions with applications to order statistics. Statistics and Probability Letters, 58, 381–388.CrossRefGoogle Scholar
Kochar, S.C. (1997). Dispersive ordering of order statistics. Unpublished manuscript.Google Scholar
Luce, R.D. (1986). Response times: Their role in inferring elementary mental organization. New York: Oxford University Press.Google Scholar
Miller, J. (1982). Divided attention: Evidence for coactivation with redundant signals. Cognitive Psychology, 14, 247–279.CrossRefGoogle ScholarPubMed
Mordkoff, J.T., Yantis, S. (1991). An interactive race model of divided attention. Journal of Experimental Psychology, 17, 520–538.Google ScholarPubMed
Moriguti, S. (1951). Extremal properties of extreme value distributions. Annals of Mathematical Statistics, 22, 523–536.CrossRefGoogle Scholar
Moriguti, S. (1953). A modification of Schwarz's inequality with applications to distributions. Annals of Mathematical Statistics, 24, 107–113.CrossRefGoogle Scholar
Munoz-Perez, J. (1990). Dispersive ordering by the spread function. Statistics & Probability Letters, 10, 407–410.CrossRefGoogle Scholar
Schweickert, R., Giorgini, M., Dzhafarov, E. (2000). Selective influence and response time cumulative distributed functions in serial-parallel task networks. Journal of Mathematical Psychology, 44, 504–535.CrossRefGoogle ScholarPubMed
Shaked, M., Shanthikumar, J.G. (1994). Stochastic orders and their applications. San Diego, CA: Academic Press.Google Scholar
Sternberg, S. (1966). High-speed scanning in human memory. Science, 153, 652–654.CrossRefGoogle ScholarPubMed
Townsend, J.T. (1990a). A potpourri of ingredients for Horse (Race) Soup. Technical Report #32. Cognitive Science Program. Indiana University, Bloomington, IN.Google Scholar
Townsend, J.T. (1990b). Serial vs. parallel processing: Sometimes they look like Tweedledum and Tweedledee but they can (and should) be distinguished. Psychological Science, 1, 46–54.CrossRefGoogle Scholar
Townsend, J.T. (1990c). The truth and consequences of ordinal differences in statistical distributions: Toward a theory of hierarchical inference. Psychological Bulletin, 108, 551–567.CrossRefGoogle Scholar
Townsend, J.T. (2001). A clarification of self-terminating vs. exhaustive variances in serial and parallel models. Perception & Psychophysics, 63, 1101–1106.CrossRefGoogle Scholar
Townsend, J.T., Ashby, F.G. (1983). The stochastic modeling of elementary psychological processes. Cambridge, MA: Cambridge University Press.Google Scholar
Townsend, J.T., Nozawa, G. (1995). Spatio-temporal properties of elementary perception: An investigation of parallel, serial, and coactive theories. Journal of Mathematical Psychology, 39, 321–359.CrossRefGoogle Scholar
Townsend, J.T., Nozawa, G. (1997). Serial exhaustive models can violate the race model inequality: Implications for architecture and capacity. Psychological Review, 104, 595–602.CrossRefGoogle Scholar
Townsend, J.T., Wenger, M.J. (2004). A theory of interactive processing: New capacity measures and predictions for a response time inequality series. Psychological Review, 111, 1003–1035.CrossRefGoogle ScholarPubMed
Van Zandt, T. (2002). Analysis of response time distributions. In Wixted, J.T. (Ed.) Stevens handbook of experimental psychology (3rd ed.). Methodology in Experimental Psychology (Vol. 4, pp. 461–516) (H. Pashler, ed.). New York: Wiley.Google Scholar
Warburton, C.E. Jr. (1967). On the application of the double exponential and Weibull distributions to failure data. Texfile Research Journal, 37, 703–704.CrossRefGoogle Scholar