The Annals of Statistics

Schur Functions in Statistics II. Stochastic Majorization

S. E. Nevius, F. Proschan, and J. Sethuraman

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This is Part II of a two-part paper. The main purpose of this two-part paper is (a) to develop new concepts and techniques in the theory of majorization and Schur functions, and (b) to obtain fruitful applications in probability and statistics. In Part II we introduce a stochastic version of majorization, develop its properties, and obtain multivariate applications of both the preservation theorem of Part I and the new notion of stochastic majorization. This leads to a definition of Schur families of multivariate distributions. Generalizations are obtained of earlier results of Olkin and of Wong and Yue; in addition, new results are obtained for the multinomial, multivariate negative binomial, multivariate hypergeometric, Dirichlet, negative multivariate hypergeometric, and multivariate logarithmic series distributions.

Article information

Ann. Statist., Volume 5, Number 2 (1977), 263-273.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 26A51: Convexity, generalizations
Secondary: 26A86 60E05: Distributions: general theory 62H99: None of the above, but in this section

Majorization Schur-concave Schur-convex Schur function stochastic majorization multivariate distributions inequalities


Nevius, S. E.; Proschan, F.; Sethuraman, J. Schur Functions in Statistics II. Stochastic Majorization. Ann. Statist. 5 (1977), no. 2, 263--273. doi:10.1214/aos/1176343793.

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See also

  • Part I: F. Proschan, J. Sethuraman. Schur Functions in Statistics I. The Preservation Theorem. Ann. Statist., Volume 5, Number 2 (1977), 256--262.