Improved Eaton bounds for linear combinations of bounded random variables, with statistical applications

The problem of evaluating tail probabilities for linear combinations of independent, possibly nonidentically distributed, bounded random variables arises in various statistical contexts, mainly connected with nonparametric inference. A remarkable inequality on such tail probabilities has been established by Eaton. The significance of Eaton’s inequality is substantiated by a recent result of Pinelis showing that the minimum BEPof Eaton’s bound BEand a traditional Chebyshev bound yields an inequality that is optimal within a fairly general class of bounds. Eaton’s bound, however, is not directly operational, because it is not explicit; apparently, it never has been studied numerically, and its many potential statistical applications have not yet been considered. A simpler inequality recently proposed by Edelman for linear combinations of iid Bernoulli variables is also considered, but it appears considerably less tight than Eaton’s original bound. This article has three main objectives. First, we put Eaton’s exact bound BEinto numerically tractable form and tabulate it, along with BEP, which makes them readily applicable; the resulting conservative critical values are provided for standard significance levels. Second, we show how further improvement can be obtained over the Eaton-Pinelis bound BEPif the number n of independent variables in the linear combination under study is taken into account. The resulting improved Eaton bounds B+EPand the corresponding conservative critical values are also tabulated for standard significance levels and most empirically relevant values of n. Finally, various statistical applications are discussed: permutation t tests against location shifts, permutation t tests against regression or trend, permutation tests against serial correlation, and linear signed rank tests against various alternatives, all in the presence of possibly nonidentically distributed (e.g. heteroscedastic) data. For permutation t tests and linear signed rank tests, the improved Eaton bounds are compared numerically with other available bounds. The results indicate that the sharpened Eaton bounds often yield sizable improvements over all other bounds considered. © 1993 Taylor & Francis Group, LLC.SCOPUS: ar.j