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June, 1982 Rectangular and Elliptical Probability Inequalities for Schur-Concave Random Variables
Y. L. Tong
Ann. Statist. 10(2): 637-642 (June, 1982). DOI: 10.1214/aos/1176345807

Abstract

It is shown that if the density $f(\mathbf{x})$ of $\mathbf{X} = (X_1, \cdots, X_n)$ is Schur-concave, then (1) $P(|X_i| \leq a_i, i = 1, \cdots, n)$ is a Schur-concave function of $(\phi(a_1), \cdots, \phi(a_n))$, and (2) $P\{\Sigma(X_i/a_i)^2 \leq 1\}$ is a Schur-concave function of $(\phi(a^2_1), \cdots, \phi(a^2_n))$, where $\phi; \lbrack 0, \infty) \rightarrow \lbrack 0, \infty)$ is any increasing and convex function. By letting $\phi(a) = a$, (1) implies that $P(|X_i| \leq a_i, i = 1, \cdots, n) \leq P(|X_i| \leq \bar{a}, i = 1, \cdots, n)$. As special consequences, the results yield bounds for exchangeable normal and $t$ variables and for linear combinations of central and noncentral Chi squared variables.

Citation

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Y. L. Tong. "Rectangular and Elliptical Probability Inequalities for Schur-Concave Random Variables." Ann. Statist. 10 (2) 637 - 642, June, 1982. https://doi.org/10.1214/aos/1176345807

Information

Published: June, 1982
First available in Project Euclid: 12 April 2007

zbMATH: 0489.62058
MathSciNet: MR653541
Digital Object Identifier: 10.1214/aos/1176345807

Subjects:
Primary: 62H99
Secondary: 26D15, 60E15

Rights: Copyright © 1982 Institute of Mathematical Statistics

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Vol.10 • No. 2 • June, 1982
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