Majorization: here, there and everywhere. (English) Zbl 1246.01010

Arnold, Barry C.

Stat. Sci. 22, No. 3, 407-413 (2007).

Summary: The appearance of A. W. Marshall and I. Olkin’s book “Inequalities: theory of majorization and its applications” [Mathematics in Science and Engineering 143; New York etc.: Academic Press (1979; Zbl 0437.26007)] on inequalities with special emphasis on majorization generated a surge of interest in potential applications of majorization and Schur convexity in a broad spectrum of fields. After 25 years this continues to be the case. The present article presents a sampling of the diverse areas in which majorization has been found to be useful in the past 25 years.

MSC:

01A60	History of mathematics in the 20th century
62-03	History of statistics

Keywords:

inequalities; Schur convex; covering; waiting time; paired comparisons; phase type; catchability; disease transmission; apportionment; statistical mechanics; random graph

Citations:

Zbl 0437.26007

References:

[1]	Aldous, D. and Shepp, L. (1987). The least variable phase type distribution is Erlang. Comm. Statist. Stochastic Models 3 467–473. · Zbl 0635.60086 · doi:10.1080/15326348708807067
[2]	Arnold, B. C. (1987). Majorization and the Lorenz Order : A Brief Introduction . Springer, Berlin. · Zbl 0649.62041
[3]	Balinski, M. L. and Young, H. P. (2001). Fair Representation Meeting the Idea of One Man , One Vote , 2nd ed. Brookings Institute Press, Washington, DC.
[4]	Dalal, S. R. and Fortini, P. (1982). An inequality comparing sums and maxima with application to Behrens–Fisher type problem. Ann. Statist. 10 297–301. · Zbl 0481.62017 · doi:10.1214/aos/1176345712
[5]	Eisenberg, B. (1991). The effect of variable infectivity on the risk of HIV infection. Statist. Medicine 9 131–139.
[6]	Hardy, G. H., Littlewood, J. E. and Pólya, G. (1929). Some simple inequalities satisfied by convex functions. Messenger of Mathematics 58 145–152. · JFM 55.0740.04
[7]	Hardy, G. H., Littlewood, J. E. and Pólya, G. (1934). Inequalities . Cambridge Univ. Press. · Zbl 0010.10703
[8]	Huffer, F. W. and Shepp, L. A. (1987). On the probability of covering the circle by random arcs. J. Appl. Probab. 24 422–429. JSTOR: · Zbl 0623.60018 · doi:10.2307/3214266
[9]	Joe, H. (1988). Majorization, entropy and paired comparisons. Ann. Statist. 16 915–925. · Zbl 0721.62067 · doi:10.1214/aos/1176350843
[10]	Marshall, A. W. and Olkin, I. (1979). Inequalities : Theory of Majorization and Its Applications . Academic Press, New York. · Zbl 0437.26007
[11]	Marshall, A. W., Olkin, I. and Pukelsheim, F. (2002). A majorization comparison of apportionment methods in proportional representation. Soc. Choice Welf. 19 885–900. · Zbl 1072.91533 · doi:10.1007/s003550200164
[12]	Nayak, T. K. and Christman, M. C. (1992). Effect of unequal catchability on estimates of the number of classes in a population. Scand. J. Statist. 19 281–287. · Zbl 0754.62086
[13]	Neuts, M. F. (1975). Computational uses of the method of phases in the theory of queues. Comput. Math. Appl. 1 151–166. · Zbl 0338.60064 · doi:10.1016/0898-1221(75)90015-2
[14]	O’Cinneide, C. A. (1991). Phase-type distributions and majorization. Ann. Appl. Probab. 1 219–227. · Zbl 0729.60069 · doi:10.1214/aoap/1177005935
[15]	Ross, S. (1981). A random graph. J. Appl. Probab. 18 309–315. JSTOR: · Zbl 0448.60010 · doi:10.2307/3213194
[16]	Ross, S. (1999). The mean waiting time for a pattern. Probab. Engrg. Inform. Sci. 13 1–9. · Zbl 0985.60081 · doi:10.1017/S0269964899131012
[17]	Stevens, W. L. (1939). Solution to a geometrical problem in probability. Ann. Eugenics 9 315–320. · Zbl 0023.05603 · doi:10.1111/j.1469-1809.1939.tb02216.x
[18]	Tong, Y. L. (1997). Some majorization orderings of heterogeneity in a class of epidemics. J. Appl. Probab. 34 84–93. JSTOR: · Zbl 0876.60012 · doi:10.2307/3215177
[19]	Zylka, C. (1985). A note on the attainability of states by equalizing processes. Theor. Chim. Acta 68 363–377.