Some New Inequalities for the Range Distribution, with Application to the Determination of Optimum Significance Levels of Multiple Range Tests

Abstract A problem arising in multiple range tests is the monotonicity of the defining critical values, which ws first treated by Lehmann and Shaffer (1977). This article proves a general inequality for the distribution function of the range, which guarantees the monotonicity of a special set of critical values. It also discusses the relation to some other inequalities. The importance of these inequalities is pointed out in considering some concepts of admissibility and optimality of multiple range tests.