The Annals of Statistics

Schur Functions in Statistics I. The Preservation Theorem

F. Proschan and J. Sethuraman

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Abstract

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

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

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176343792

Digital Object Identifier
doi:10.1214/aos/1176343792

Mathematical Reviews number (MathSciNet)
MR443224

Zentralblatt MATH identifier
0361.62055

JSTOR
links.jstor.org

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

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

Citation

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