## The Annals of Statistics

- Ann. Statist.
- Volume 10, Number 1 (1982), 297-301.

### An Inequality Comparing Sums and Maxima with Application to Behrens-Fisher Type Problem

Siddhartha R. Dalal and Peter Fortini

#### 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.

#### Article information

**Source**

Ann. Statist., Volume 10, Number 1 (1982), 297-301.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176345712

**Digital Object Identifier**

doi:10.1214/aos/1176345712

**Mathematical Reviews number (MathSciNet)**

MR642741

**Zentralblatt MATH identifier**

0481.62017

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62F25: Tolerance and confidence regions

Secondary: 60E15: Inequalities; stochastic orderings

**Keywords**

Majorization inequality sums of powers multiple comparisons Behrens-Fisher problem

#### Citation

Dalal, Siddhartha R.; Fortini, Peter. An Inequality Comparing Sums and Maxima with Application to Behrens-Fisher Type Problem. Ann. Statist. 10 (1982), no. 1, 297--301. doi:10.1214/aos/1176345712. https://projecteuclid.org/euclid.aos/1176345712