The Annals of Statistics

Schur Functions in Statistics II. Stochastic Majorization

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

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Abstract

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

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

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176343793

Digital Object Identifier
doi:10.1214/aos/1176343793

Mathematical Reviews number (MathSciNet)
MR443225

Zentralblatt MATH identifier
0383.62039

JSTOR
links.jstor.org

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

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

Citation

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. https://projecteuclid.org/euclid.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.