×

Multivariate arrangement increasing functions with applications in probability and statistics. (English) Zbl 0648.62052

Summary: A real valued function of s vector arguments in \(R^ n\) is said to be arrangement increasing if the function increases in value as the components of the vector arguments become more similarly arranged. Various examples of arrangement increasing functions are given including many joint multivariate densities, measures of concordance between judges and the permanent of a matrix with nonnegative components. Preservation properties of the class of arrangement increasing functions are examined, and applications are given including useful probabilistic inequalities for linear combinations of exchangeable random vectors.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
60D05 Geometric probability and stochastic geometry
60E15 Inequalities; stochastic orderings
62H20 Measures of association (correlation, canonical correlation, etc.)
26D15 Inequalities for sums, series and integrals
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Boland, P. J.; Proschan, F.; Tong, Y. L., Moment and geometric probability inequalities arising from arrangement increasing functions, Ann. Probab (1987), in press · Zbl 0638.60013
[2] Derman, C.; Lieberman, G.; Ross, S., On optimal assembly of systems, Naval. Res. Logist. Quart., 19, 569-574 (1972) · Zbl 0251.90019
[3] Diaconis, P.; Graham, R. L., Spearman’s footrule as a measure of disarray, J. Roy. Statist. Soc. Ser. B, 39, 262-268 (1977) · Zbl 0375.62045
[4] Ehrenberg, A. S.C, On sampling from a population of rankers, Biometrika, 39, 82-87 (1952) · Zbl 0046.35901
[5] Hays, W. L., A note on average tau as a measure of concordance, J. Amer. Statist. Assoc., 331-341 (1960) · Zbl 0212.22403
[6] Hollander, M.; Proschan, F.; Sethuraman, J., Functions decreasing in transposition and their applications in ranking problems, Ann. Statist., 5, 722-733 (1977) · Zbl 0356.62043
[7] Kendall, M. G., (Rank Correlation Methods (1970), Hafner: Hafner New York) · Zbl 0199.53501
[8] Lorentz, G. G., An inequality for rearrangements, Amer. Math. Monthly, 60, 176-179 (1953) · Zbl 0050.28201
[9] Marshall, A.; Olkin, I., (Inequalities: Theory of Majorization and Its Applications (1979), Academic Press: Academic Press New York) · Zbl 0437.26007
[10] Minc, H., Rearrangements, Trans. Amer. Math. Soc., 159, 497-504 (1971) · Zbl 0225.26020
[11] Ruderman, H. D., Two new inequalities, Amer. Math. Monthly, 59, 29-32 (1952) · Zbl 0046.05101
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.