Directed Alternatives in Testing

Abstract

Abstract Consider a hypothesis‐testing problem concerned with a one‐dimensional parameter θ. In testing a null hypothesis H : θ = θ o , a two‐sided alternative is K : θ ≠ θ o . A one‐sided alternative is K : θ > θ o . This latter alternative is appropriate when it is reasonably certain that θ cannot be less then θ o . The advantage of posing a one‐sided alternative is that the appropriate one‐sided test is much more powerful than the corresponding two‐sided test. The idea of one‐sided and two‐sided alternatives extends to the case of testing a k ‐dimensional parameter θ . If the null hypothesis is H : θ = θ o , then the analogue to the two‐sided alternative is K : θ ≠ θ o . This latter alternative is said to be unrestricted or not directed. An example of the analogue to the one‐sided alternative is K : θ i ≥ θ io , for i = 1, …, k , with strict inequality for at least one index i . Here θ = (θ 1 , …θ k ). Such an alternative is called a directed alternative . We introduce the notion of directed alternatives for multidimensional parameters. Among the special cases treated are the simple order alternative in which parameters are nondecreasing, the tree order alternative in which case one mean parameter is assumed to be less than or equal to each of the other mean parameters, and the stochastic order alternative in which one multinomial distribution is more concentrated at higher categories than another distribution. We indicate appropriate tests for these directed alternatives.