More powerful tests for the sign testing problem about gamma scale parameters

For i=1, … , p, let (Formula presented.) denote independent random samples from gamma distributions with unknown scale parameters θ_i and known shape parameters η_i. Consider testing H₀:θ_i≤θ_i0 for some i=1, … , p versus H₁:θ_i>θ_i0 for all i=1, … , p, where θ₁₀, … , θ_p0 are fixed constants. For any 0<α<0.4, we construct two new tests that have the same size as the likelihood ratio test (LRT) and are uniformly more powerful than it. Power comparisons of our tests with other tests are given. The proposed tests are intersection–union tests. We apply the results to test the variances of normal distributions and scale parameters of two-parameter exponential distributions. Finally, we illustrate our proposed tests with an example.