More powerful tests for the sign testing problem

For i = 1,..., p, let X_i1,..., X_ini denote independent random samples from normal populations. The ith population has unknown mean μ_i and unknown variance σ_i². We consider the sign testing problem of testing H₀: μ_i ≤ a_i, for some i=1,..., p, versus H₁: μ_i > a_i, for all i=1,..., p, where a₁,..., a_p are fixed constants. Here, H₁ might represent p different standards that a product must meet before it is considered acceptable. For 0 < α < 1/2, we first derive the size-α likelihood ratio test (LRT) for this problem, and then we describe an intersection-union test (IUT) that is uniformly more powerful than the likelihood ratio test if the sample sizes are not all equal. For a more general model than the normal, we describe two intersection-union tests that maximize the size of the rejection region formed by intersection. Applying these tests to the normal problem yields two tests that are uniformly more powerful than both the LRT and IUT described above. A small power comparison of these tests is given.