The Annals of Statistics

Schur Functions in Statistics I. The Preservation Theorem

F. Proschan and J. Sethuraman

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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.

Article information

Ann. Statist., Volume 5, Number 2 (1977), 256-262.

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62H99: None of the above, but in this section
Secondary: 26A86

Majorization Schur-concave Schur-convex integral transformation shock models moment inequalities stochastic majorization multivariate distributions


Proschan, F.; Sethuraman, J. Schur Functions in Statistics I. The Preservation Theorem. Ann. Statist. 5 (1977), no. 2, 256--262. doi:10.1214/aos/1176343792.

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See also

  • Part II: S. E. Nevius, F. Proschan, J. Sethuraman. Schur Functions in Statistics II. Stochastic Majorization. Ann. Statist., Volume 5, Number 2 (1977), 263--273.