The Annals of Statistics

Rectangular and Elliptical Probability Inequalities for Schur-Concave Random Variables

Y. L. Tong

Full-text: Open access

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.

Article information

Source
Ann. Statist., Volume 10, Number 2 (1982), 637-642.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176345807

Digital Object Identifier
doi:10.1214/aos/1176345807

Mathematical Reviews number (MathSciNet)
MR653541

Zentralblatt MATH identifier
0489.62058

JSTOR
links.jstor.org

Subjects
Primary: 62H99: None of the above, but in this section
Secondary: 26D15: Inequalities for sums, series and integrals 60E15: Inequalities; stochastic orderings

Keywords
Probability inequalities in multivariate distributions Schur-functions and majorization bounds for multivariate normal and $t$ probabilities bound for central and noncentral Chi squared probabilities

Citation

Tong, Y. L. Rectangular and Elliptical Probability Inequalities for Schur-Concave Random Variables. Ann. Statist. 10 (1982), no. 2, 637--642. doi:10.1214/aos/1176345807. https://projecteuclid.org/euclid.aos/1176345807


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