A class of k-sample distribution-free tests for the umbrella alternatives

We propose a class of k-sample distribution-free tests for the umbrella location alternatives when the peak of the umbrella, 1 is known. The proposed class of tests is the sum of 1-sample and (k - l)-sample tests. These tests are weighted consecutive linear combinations of two-sample U-statistics of order (b, d) whose kernel is a function of sub-sample medians. The rank expressions of the proposed class and its performance in terms of Pitman asymptotic relative efficiency (ARE) for optimal weights are given.