Multivariate generalisations of k-sample rank tests for umbrella alternatives

Abstract

We consider the problem of multivariate generalisation of k-sample rank tests for umbrella alternatives. That is, testing the null hypothesis of equality of medians against the alternatives of interest of the form for all x and g=1, …, p; with at least one strict inequality for at least one g when ℓ is known. This is known as umbrella alternatives and from now on ℓ will be referred to as the peak of the umbrella. Sometimes when comparing populations, we are interested in whether the medians of the groups are increasing up to some point then decreasing as the treatment levels are increasing. We propose a multivariate generalisation of umbrella alternatives based on coordinate-wise approach when the peak of the umbrella ℓ is known. Approximate critical values for the small sample null distributions are also discussed. Furthermore, we study the power of the test under location-family alternatives using small sample simulations.