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March, 1977 Schur Functions in Statistics I. The Preservation Theorem
F. Proschan, J. Sethuraman
Ann. Statist. 5(2): 256-262 (March, 1977). DOI: 10.1214/aos/1176343792


This is Part I of a two-part paper; the 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. The main theorem of Part I states that if $f(x_1, \cdots, x_n)$ is Schur-concave, and if $\phi(\lambda, x)$ is totally positive of order 2 and satisfies the semigroup property for $\lambda_1 > 0, \lambda_2 > 0: \phi(\lambda_1 + \lambda_2, y) = \int \phi(\lambda_1, x)\phi(\lambda_2, y - x) d\mu(x)$, where $\mu$ is Lebesgue measure on $\lbrack 0, \infty)$ or counting measure on $\{0, 1, 2, \cdots\}$, then $h(\lambda_1, \cdots, \lambda_n) \equiv \int \cdots \int \Pi^n_1 \phi(\lambda_i, x_i)f(x_1, \cdots, x_n) d\mu(x_1) \cdots d\mu(x_n)$ is also Schur-concave. This theorem is then applied to obtain renewal theory results, moment inequalities, and shock model properties.


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F. Proschan. J. Sethuraman. "Schur Functions in Statistics I. The Preservation Theorem." Ann. Statist. 5 (2) 256 - 262, March, 1977.


Published: March, 1977
First available in Project Euclid: 12 April 2007

zbMATH: 0361.62055
MathSciNet: MR443224
Digital Object Identifier: 10.1214/aos/1176343792

Primary: 62H99
Secondary: 26A86

Keywords: integral transformation , majorization , Moment inequalities , Multivariate distributions , Schur-concave , Schur-convex , shock models , stochastic majorization

Rights: Copyright © 1977 Institute of Mathematical Statistics

Vol.5 • No. 2 • March, 1977
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