Two known results in multiple testing, one relating to the directional error control of augmented step-down procedure proved by Shaffer (1980) and the other on the monotonicity of the critical values of step-up procedure proved by Dalal and Mallows (1992) , are revisited and given alternative proofs in this article. 1. Introduction. Testing of a null hypothesis against two-sided alternative is typically considered as a problem of making one of two kinds of decision, acceptance or rejection of

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... e null hypothesis, and is designed in such a way that the Type I error rate associated with false rejection of the null hypothesis is controlled at a specified value. Once the null hypothesis is rejected, the direction of the alternative hypothesis is decided based on the value of the test statistic. However, a directional error or Type III error might occur in making such directional decisions. For instance, in testing H 0 : θ = θ 0 against H 1 : θ = θ 0 , with θ being the parameter of a random variable T and θ 0 being some known value, a rejection region of the form T ≤ a or ≥ b is used, where a and b are determined subject to a specified control of the Type I error rate, i.e., the probability of falsely rejecting H 0 . Once H 0 is rejected, the decision regarding θ > θ 0 or θ < θ 0 is made by checking if T ≥ b or T ≤ a. A Type III error occurs when, for example, θ < θ 0 (or θ > θ 0 ) is the true situation, but we falsely decide for θ > θ 0 (or θ < θ 0 ) after rejection of H 0 . It is interesting to see, however, that in almost all testing situations where T stochastically increases with θ, controlling the Type I error rate will ensure the same control for the Type III error rate. This is because, when θ = θ 0 , there is no Type III error. On the other hand, when θ < θ 0 , the chance of Type III error, which is