LetX1, …,Xkbekstochastically independent random variables and letXihave the uniform distribution on the interval from 0 to θi,i= 1, …,k. The power function of the likelihood ratio test is found for the problem of testing {H0: θ1= … = θkθ0, θ0specified;H1: θi≠ θ0for at least onei,i= 1, …,k}, and ump possibilities are investigated. For the problem of testing {H0: θ1= θ2, common value unspecified;H1: θ1≠ θ2}, the power function of the likelihood ratio test is found and this test is shown to be an invariant test under multiplication by a strictly positive constant and to be a ump unbiased test. It is pointed out that when sample sizes are equal, the likelihood ratio test statistic is distributed as the absolute value of a certain Laplace random variable, and that the likelihood ratio test is ump invariant under permutations and multiplication by a strictly positive constant. An iterative test for the problem of testing {H0: θ1= … = θk, common value unspecified;H1: θi≠ θjfor at least one (i,j),i≠j,i,j= 1,…,k},k> 2, is discussed. These results are extended to a larger class of distributions.