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

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.