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March, 1982 An Inequality Comparing Sums and Maxima with Application to Behrens-Fisher Type Problem
Siddhartha R. Dalal, Peter Fortini
Ann. Statist. 10(1): 297-301 (March, 1982). DOI: 10.1214/aos/1176345712

Abstract

A sharp inequality comparing the probability content of the $\ell_1$ ball and that of $\ell_\infty$ ball of the same volume is proved. The result is generalized to bound the probability content of the $\ell_p$ ball for arbitrary $p \geq 1$. Examples of the type of bound include $P\{(|X_1|^p + |X_2|^p)^{1/p} \leq c\} \geq F^2(c/2^{1/2p}),\quad p \geq 1,$ where $X_1, X_2$ are independent each with distribution function $F$. Applications to multiple comparisons in Behrens-Fisher setting are discussed. Multivariate generalizations and generalizations to non-independent and non-exchangeable distributions are also discussed. In the process a majorization result giving the stochastic ordering between $\Sigma a_i X_i$ and $\Sigma b_i X_i$, when $(a^2_1, a^2_2, \cdots, a^2_n)$ majorizes $(b^2_1, b^2_2, \cdots, b^2_n)$, is also proved.

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Siddhartha R. Dalal. Peter Fortini. "An Inequality Comparing Sums and Maxima with Application to Behrens-Fisher Type Problem." Ann. Statist. 10 (1) 297 - 301, March, 1982. https://doi.org/10.1214/aos/1176345712

Information

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

zbMATH: 0481.62017
MathSciNet: MR642741
Digital Object Identifier: 10.1214/aos/1176345712

Subjects:
Primary: 62F25
Secondary: 60E15

Rights: Copyright © 1982 Institute of Mathematical Statistics

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Vol.10 • No. 1 • March, 1982
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